Download Chapter 7 Fluorescence Imaging of Quantum Gases

February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
ChapterWeitenberg
Chapter 7
Fluorescence Imaging of Quantum Gases
Christof Weitenberg
Institut für Laser-Physik, Universität Hamburg,
Luruper Chaussee 149, D-22761 Hamburg, Germany
Quantum gases in optical lattices have proven a prolific platform to
study condensed matter models such as the Bose Hubbard model. The
recently achieved in situ fluorescence imaging of low-dimensional systems has pushed the detection capabilities to a fully microscopic level.
The method yields single-site and single-atom resolved images of the
lattice gas in a single experimental run, thus giving direct access to
fluctuations and correlation functions in the many-body system. These
quantum gas microscopes have been used to study the superfluid Mott
insulator quantum phase transition at the single-atom level. Moreover,
single-site resolved addressing allows flipping the spin of individual atoms
in a Mott insulator, thus deterministically creating local spin excitations
whose dynamics can be observed. In this chapter, we will describe the
implementation of the technique and discuss some of the obtained results.
7.1. Introduction
Fluorescence imaging is a means of obtaining a large signal from a single
atom by holding it in place while detecting the photons scattered from an
illuminating resonant laser beam. It has been applied to the detection of
single ions in Paul traps1 and single atoms in dipole traps2 with high fidelity. The application of fluorescence imaging to atoms in an optical lattice
has first been demonstrated in a regime, where the atoms were optically
resolvable, be it due to sparse filling3 or a large lattice spacing.4 An alternative approach is scanning electron microscopy,5 which reaches impressive
resolution, but so far no full single-atom sensitivity.
Recently, experiments at Harvard and at the Max-Planck-Institute for
Quantum Optics (MPQ) have pushed fluorescence imaging to the Bose
1
February 16, 2014
2
0:15
World Scientific Review Volume - 9in x 6in
ChapterWeitenberg
Christof Weitenberg
Hubbard regime, which requires high resolution of cold dense samples at
small lattice spacings.6–8 This advance opens up the possibility of studying
many-body physics at the fundamental level of individual atoms. The Bose
Hubbard Hamiltonian incorporates a tunneling matrix element J and an
on-site interaction energy U , whose ratio can be tuned via the lattice depth
V . The competition between these two energies gives rise to a quantum
phase transition between a superfluid phase for weak interactions and a
Mott insulating phase for strong interactions (see Chapter 3).
The Bose Hubbard physics was previously studied with cold atoms.9
The phase coherence between the lattice sites that is present in the superfluid state was directly revealed in the interference pattern after a free
expansion of the atoms and the vanishing of this interference for increasing interactions was the first indication for the Mott insulating phase.10
The complementary observable is the number squeezing in the Mott insulator state, which leads to the formation of plateaus of integer filling in the
density profile, as has been observed via high-resolution in situ absorption
imaging.11
This is where single-atom resolved fluorescence imaging steps in and
finds its natural playground. The sensitivity to single atoms makes it an
ideal probe for the number fluctuations that are at the heart of these manybody states. One limitation of the current fluorescence experiments is that
they can only detect the parity of the occupation number on a lattice site,
because the atoms get lost pairwise in light-assisted collisions. However,
in many cases this restricted observable still allows deep insight into the
many-body state under study.
The on-site parity yields information about the density profile and also
the number fluctuations. A density of one atom per site is only expected
for the perfect number squeezing deep in the Mott insulator, because any
thermal or quantum fluctuation will lower the average density after parity projection. The reduction of number fluctuations across the superfluid
Mott insulator transition was directly verified via fluorescence imaging.7
Deep in the Mott insulating phase the remaining fluctuations are thermally activated and therefore the density profile of the trapped system is
a good measure for the temperature of the lattice system.8 The resulting
precise thermometry was recently used for lattice modulation spectroscopy
(compare Chapters 10, 11) to reveal the Higgs mode close to the superfluid
Mott insulator transition.12 Lattice amplitude modulation was also used to
demonstrate an orbital excitation blockade13 and photon-assisted tunneling.14
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
Fluorescence Imaging of Quantum Gases
ChapterWeitenberg
3
Fluorescence detection yields a single-atom resolved image in a single
experimental run. Therefore, it is possible to extract correlations between
different lattice sites. Two-site parity parity correlation functions were
studied across the superfluid Mott insulator transitions in 2D and 1D systems.15 The study was also extended to three-site correlations and to a
non-local string order parameter in the Mott insulating phase.15 The timeresolved detection of propagating correlations after a sudden change of the
Hamiltonian parameters revealed light-cone-like spreading with a finite velocity.16 The general approach is not restricted to Hubbard-type physics,
as was recently demonstrated by the detection of correlations between Rydberg atoms forming spatially ordered structures.17
The high-resolution technique can also be used to control the trapping
potential on the scale of the lattice spacing18 or to flip the spin of individual
atoms in a Mott insulator.19 This technique has recently been employed to
study the dynamics of a single spin impurity20 and of magnon-bound states
of two impurities21 in a strongly interacting 1D system. While here the spin
was encoded in two atomic hyperfine states,22 one can alternatively identify
the spin with the site-occupation in a tilted lattice, yielding stronger spin
interactions.23 This mapping was used to simulate a chain of interacting
quantum Ising spins and to study the phase transition from a paramagnetic
to an antiferromagnetic phase.24
In this chapter, we will describe the technical details of the fluorescence
imaging method, the main challenges and possible solutions to them. Then
we will discuss examples of the measurement of the occupation numbers and
the two-site correlation functions in the context of Bose Hubbard physics.
For the spin physics, we will focus on the mapping to hyperfine states,
presenting the manipulation of individual spins and the measurement of
the ensuing dynamics.
7.2. Experimental realization
7.2.1. High-resolution imaging system
The experimental realization of single-site resolved imaging requires exceptionally high resolution over a large field of view, which is a very challenging
demand considering the limited optical access in a quantum gas machine.
The resolution R of an imaging system with a numerical aperture NA is
given by R = 0.61λ/NA, where the imaging wavelength λ is fixed by the
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
4
ChapterWeitenberg
Christof Weitenberg
(a)
(c)
5 µm
0.6 µm
(b)
z
(d)
5 µm
y
x
y
x
Fig. 7.1. Experimental setups of quantum gas microscopes. (a) Setup of the MPQ
experiment.8 An ultracold sample of about 1000 87 Rb atoms is prepared in a single
antinode of a vertical lattice, produced by a laser beam at 1064 nm which is retroreflected from the vacuum window. Additional lattices in the horizontal plane are added
to enter the Bose Hubbard regime (red arrows). All lattices are ramped up to freeze the
distribution while it is illuminated by an optical molasses superimposed with the lattice
beams. A high-resolution objective is placed outside the vacuum chamber and images
the atoms with a resolution of 700 nm. (d) Setup of the Harvard experiment.6 The
87 Rb atoms are trapped in a 2D surface trap a few micrometers below the lower surface
of a hemispherical lens inside the vacuum chamber. This lens serves to increase the
numerical aperture of the objective lens outside the vacuum by the index of refraction,
from NA=0.55 to NA=0.8. The atoms are illuminated from the side by the molasses
beams (red arrows). The lattice is created by projecting a periodic phase mask onto
the atoms through the objective. Light with a very short coherence length of 100 µm is
used to suppress interferences with stray light. (b),(c) Corresponding false color fluorescence images of sparse clouds. Subfigures (a) and (b) taken from Ref. 8. Subfigure (c)
adapted by permission from Macmillan Publishers Ltd: Nature 462, 74–77, copyright
2009. Subfigure (d) with kind permission from Markus Greiner.
atomic resonance (λ = 780 nm for rubidium atoms)a . To reach the Bose
Hubbard regime with a sufficient tunnel coupling J, the lattice spacing alat
must be on the order of 500 nm (532 nm in the MPQ experiment and 680 nm
in the Harvard experiment). For R ≈ alat to hold, this poses stringent requirements on the NA.
The MPQ experiment used a customized commercial objective with
NA = 0.68 and a working distance of 13 mm, positioned just outside the
vacuum chamber [see Fig. 7.1(a)]. The Harvard experiment positioned a
first hemispheric lens inside the vacuum cell and placed the atoms a few
a Using
the shorter wavelength of higher transitions25,26 that leads to higher resolution
is a promising route for future experiments.
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
Fluorescence Imaging of Quantum Gases
ChapterWeitenberg
5
micrometers below, thus increasing the NA = 0.55 of the objective outside
the vacuum to NA = 0.8 [see Fig. 7.1(d)]. The respective resolutions in the
two experiments were 700 nm and 600 nm.
Great care has to be taken to avoid aberrations that compromise the
resolution. The deformations of the vacuum window through which one
images must be smaller than λ/4 over the illuminated area. This can be
compromised due to stress in the glass and must be verified interferometrically in the final setup. The objective must be properly aligned perpendicular to the vacuum window. The performance of the imaging system can
be directly verified by measuring the point spread function (PSF) of single
isolated atoms in the lattice [see Fig. 7.3(e)]. A tilt of the objective will,
e.g., show immediately in an asymmetric shape of the PSF.
For such high NA imaging systems, the depth of focus is only on the
order of a micrometer. The objective therefore needs to be mounted on
a positioner that allows regular refocussing of the images, to compensate
for thermal drifts of the setup. The small depth of focus also limits the
quantum gas microscope to single 2D systems that can be prepared as
discussed in Chapter 6.
7.2.2. Fluorescence imaging
For reaching single-atom sensitivity, fluorescence imaging is favorable compared to absorption imaging, because it yields a larger signal-to-noise ratio.
The strategy is to hold the atom in place with a pinning lattice and to scatter photons for as long as one second. The atomic distribution is frozen by
ramping up the lattice on a time scale faster than the many-body dynamics. While the atoms scatter light, they have to stay in their lattice site.
Therefore they are illuminated by laser beams in an optical molasses configuration (see Section 3.5.1 for details)b that keeps them at a temperature of
about 20 µK. The pinning lattice has to be around ten times deeper than
this temperature in order to prevent thermal hopping of the atoms.4 This
is a few hundred times deeper than the typical lattice depths of some 10 Er ,
at which the Bose Hubbard physics occurs (here Er = h2 /(8ma2lat ) is the
recoil energy with the atomic mass m and Planck’s constant h).
There are different strategies to deal with this problem. The Harvard
group used two different wavelengths: a far-detuned wavelength (755 nm)
with a small heating rate for the Bose Hubbard physics and a nearoptical molasses can be either in a σ + -σ − -configuration8 or in a lin-perp-linconfiguration.6
b The
February 16, 2014
0:15
6
World Scientific Review Volume - 9in x 6in
ChapterWeitenberg
Christof Weitenberg
resonance wavelength (detuned by +50 GHz from the rubidium D1 line)
with a large light shift for pinning the atoms. Changing the wavelength
was possible without changing the geometry of the lattice, because it was
generated by projecting a periodic mask onto the atoms through the objective, such that the lattice spacing was independent of the wavelength.
In the MPQ experiment, the lattice was produced by retroreflection of the
lattice beams with a far-detuned wavelength of 1064 nm and the pinning
was achieved by changing the optical power to high values (10 W per axis).
It is important to assure that the pinning lattice does not interfere with
the proper functioning of the optical molasses. First, one has to avoid any
vector light shift in the pinning lattice, which would disturb the molasses
just like a magnetic field. This demands a detuning of the lattice light of at
least a few 10 GHz; but even at a wavelength of 1064 nm, it is important to
use a well-controlled linear polarization of the lattice beams. Second, in the
3D configuration of an optical molasses, the interference of the beams leads
to intensity modulations. At some points in space, the intensity is too low
to sustain a proper working of the molasses. As the atoms are pinned in
their lattice sites, they will not move out of these areas. Therefore the phase
of the molasses beams has to be modulated to wash out these interferences.c
A few numbers shall illustrate the operation of the optical molasses.8
Typical values of −80 MHz detuning (including the shift due to the pinning
lattice) and a total intensity of 7 times the saturation intensity lead to a
scattering rate of 150 kHz. The objective of NA = 0.68 collects 13 % of
the scattered photons. Including the transmission of the optics and the
quantum efficiency of the camera as well, in total 7 % of the photons are
detected. Using an EMCCD camera cooled down to −70o C, one obtains a
signal of 30,000 counts per atom in 1 s illumination time. This can easily
be distinguished from noise of the camera and from imperfect stray light
subtraction. Due to the very long imaging times in fluorescence imaging,
the detection fidelity can actually be limited by atom loss from collisions
with atoms from the background gas. For an illumination time of 1 s, an
atom loss of 1 % was measured, which corresponds to a trap lifetime of 75 s.
An imaging time of a few 100 ms is usually sufficient to get enough signal
and is therefore a good compromise.
c This
is achieved by a small detuning between the molasses beams and by a modulation
of the retroreflecting mirrors on the scale of 100 Hz, i.e. slow compared to the scattering
rate of the molasses. Short term interference is required if one beam is not balanced by
a counterpropagating beam (as the one from below in the MPQ experiment), because
cooling in this direction relies on the polarization gradients with the other axes. Omitting
the retroreflexes is possible, because the lattice absorbs the net momentum.
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
Fluorescence Imaging of Quantum Gases
7
(c) molasses: light-assisted collisions
V = 10 Er
(a) strongly-correlated state
ChapterWeitenberg
(d) observed occupations
V = 3000 Er
(b) pin atoms: initial occupations
Fig. 7.2. Illustration of the pinning and the parity projection. (a) Atoms (circles) in
an optical lattice (line). At a shallow lattice of around 10 Er depth, the atoms are in a
strongly correlated state and possibly delocalized over the lattice (the arrows indicate
the tunnel coupling between the sites). (b) When the lattice depth is quickly ramped
up to 3000 Er depth, the atoms are pinned to fixed occupation numbers and loose the
phase coherence. (c) During the molasses, pairs of atoms undergo light-assisted collisions
and are quickly lost from the trap before contributing to the signal (d) The observed
occupations are given by the parity of the initial occupations. One measures either zero
or one atom per site.
7.2.3. Parity projection
One important technical problem of the fluorescence imaging are the socalled light-assisted collisions.27,28 When two atoms collide in the presence
of resonant light and one atom is excited, they can gain a lot of kinetic
energy in the very steep corresponding molecular potential, leading to the
loss of both atoms from the trap. At the high densities within a lattice site,
this process is very fast and the atoms disappear before contributing to the
fluorescence signal. This pairwise loss accounts to measuring the parity of
the occupation of a lattice site (see Fig. 7.2)d .
The parity projection has implications on the analysis of the data. However, in the following sections we will see that one can do a lot of things with
just zeros and ones. In the future it might be worthwhile to circumvent the
parity projection. This can be done by working in a dilute regime where
d It
was found that the expelled atoms can be recaptured by the molasses at random
places in the lattice. To avoid this, atom pairs can be removed in a controlled way
by a single-beam 50 ms push-out pulse before switching on the molasses. The laser
beam operates on the F = 2 to F 0 = 3 transition, which is 6.8 GHz red detuned for
the rubidium atoms that are prepared in F = 1, but efficiently excites them into the
molecular potential.
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
8
ChapterWeitenberg
Christof Weitenberg
(b)
(a)
2 µm
(c)
3
1
Counts (x 1000)
2
y
x
(e)
10
Counts (x 100)
(d)
signal
model PSF
8
6
4
2
0
0
0.5
1
1.5
Radial distance r (µm)
signal (a.u.) signal (a.u.)
0
12
(f)
1 2 3 4 5 6 7
Distance (alat )
Fig. 7.3. Illustration of the image analysis. (a) Typical experimental fluorescence image
of a dense cloud. (b) Reconstructed image convoluted with the experimental point spread
function. (c) Digitized image of the occupations on the lattice. The blue dots mark the
lattice sites, the black circles indicate the presence of an atom. The reconstruction
algorithm tries different occupations on the lattice and compares the convoluted image
with the experimental image. (d) Experimental fluorescence image with sparse filling.
Images like this are used to precisely characterize the lattice. The lattice spacing and
orientation are obtained from analyzing the positions of the isolated atoms. The lattice
position is fixed by an isolated atom (the identified lattice sites are marked by white
points). (e) Radial average of the experimentally obtained PSF obtained from an average
of many isolated atoms. The red line is a fit of a double Gaussian to the data. (f)
Illustration of the required resolution. When five PSF at neighboring lattice sites (solid
lines) are added, they yield a flat signal (dashed line), so that the atoms are not resolved
(upper panel). However, when the central atom is missing, this is clearly visible in the
added signal (lower panel). Figure adapted from Refs. 8 and 29.
occupations higher than one are negligible. This regime can be reached by
working with two different lattice spacings for the physical lattice for the
Bose Hubbard physics and the pinning lattice for the freezing of the atoms.
When the pinning lattice has a smaller lattice spacing, each lattice site of
the physical lattice can be mapped to several sites of the pinning lattice.
Thus one can arrange it so that even for higher occupations in the physical lattice, each atom has its own pinning lattice site during the imaging.
Finally, one could dilute the system in the direction orthogonal to the 2D
plane via laser-assisted tunneling after freezing the distribution.
7.2.4. Image analysis
The resolution criterion R ≈ alat does not have to be strictly fulfilled for
identifying the occupations on a lattice. The additional information on the
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
Fluorescence Imaging of Quantum Gases
ChapterWeitenberg
9
lattice structure and the discrete number of atoms can be used to beat the
diffraction limit, when a good signal-to-noise ratio is given. In the case of
dilute 1D lattices, one can even enhance the resolution by a factor of four
by numerical post-processing.30 In the case of a densely filled 2D lattice,
the ratio R/alat = 1.3 is sufficient to fully reconstruct the occupations on
the lattice with very high fidelity of better than 99 % as demonstrated in
Ref. 8. Here the parity projection is an advantage, because one does not
have to discern different higher occupations.
The working of the reconstruction algorithm8 is illustrated in Fig. 7.3.
First the lattice spacing and the lattice orientation (here at about ±45o )
are determined precisely from sparse images as in Fig. 7.3(d). The absolute
position of the lattice in each image is then found by identifying isolated
atoms at the edges of the cloud. This position slowly moves from shot to
shot by a few percent of a lattice spacing due to thermal drifts of the setup.
Then an algorithm places atoms on the lattice sites, convolutes this digitized
image with the measured PSF (allowing for varying light levels ±20 %) and
compares the result with the original image. It runs an optimization scheme
on the distribution of atoms minimizing the difference between the original
and the reconstructed image. For the smaller ratio of R/alat = 0.88 of
Ref. 7, it is sufficient to set a threshold for the signal per lattice site to
decide for the absence or presence of an atom.
Once the occupations on the lattice are determined, one has noise-free
digitized images without the complications of non-linear effects and experimental systematics of absorption imaging (see Chapter 6) and one has
access to the local density and its fluctuations without further calibration
factors.
7.3. Bose Hubbard physics at the single-atom level
7.3.1. Shell structure and number fluctuations
In situ fluorescence images of ultracold atoms are a terrific tool for studying
the Bose Hubbard model. Fig. 7.4 recalls the schematic ground state phase
diagram of the Bose Hubbard model as a function of tunneling strength J/U
and chemical potential µ/U both in units of the interaction energy U (compare Fig. 1.1). The blue regions mark the Mott lobes of different fillings,
the bright grey region corresponds to the superfluid phase. In most experiments, there is an external harmonic confinement VT (r) = 21 m(ωx2 x2 +ωy2 y 2 )
on top of the lattice potential. For a sufficiently smooth confinement, the
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
10
ChapterWeitenberg
Christof Weitenberg
system locally behaves like a homogeneous system at the same temperature
and with the local chemical potential µloc (r) = µ0 − VT (r), where µ0 is the
chemical potential in the center of the trap (local density approximation; see
also Section 6.3.2).
Spatially resolved imaging allows making resolved measurements within
the phase diagram. The radial profile of the trapped system will probe a
vertical cut through the phase diagram. For example, the different Mott
lobes in the phase diagram will lead to the formation of concentric shells
with integer fillings in real space [compare the red and orange arrow in
Fig. 7.4(a) and (b)]. On the other hand, one can probe at a fixed chemical
potential by restricting the observation, e.g. to a few lattice sites in the
center of the trap. Varying the lattice depth to change the ratio U/J,
one can then study the fluctuations7 or the correlations15 across the phase
transition between the superfluid and the Mott insulator (blue arrows).
Figure 7.5(a) shows two images of Mott insulators in the zero-tunneling
limit (J/U → 0, here U/J ∼ 300) that were prepared with different total atom numbers. One clearly sees the high degree of number-squeezing
with exactly one atom per site and only few individual defects. For the
larger cloud, an n = 2 shell forms in the center, which shows here as empty
sites due to parity projection.e To analyze the images, radial atom density
and variance profiles were obtained from an azimuthal average within the
individual images [Fig. 7.5(b)]. The density profiles can be compared to a
simple grand canonical model for the zero-tunneling limit, which shows how
the shell structure is smoothened by thermal excitations that form preferably at the edges of the Mott shells.34 The initial occupation probability
for n atoms at a lattice site at radius r is given by
1 [(µloc (r)n−En )/(kB T )]
e
,
(7.1)
Z(r)
P
with the partition function Z(r) = n e[(µloc (r)n−En ])/(kB T ) and the interaction energy En = U n(n−1)/2. Taking the parity projection into account,
P
we have the detected density n̄det (r) = n mod2 (n)Pr (n), which can be fitted to the radial density profiles [Fig. 7.5(b)]. The fit gives the temperature
in units of U from a single image with just a few percent error. This precise
in situ thermometry is possible due to a good knowledge of the trapping
potential and due to the simple analytic expression of the density profile
Pr (n) =
e Up
to four shells alternating between occupations of 1 and 0 were observed in the
Harvard experiment [see Fig. 7.4(c)], well demonstrating that the mechanism of pairwise
atom loss also works for larger initial occupations.
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
ChapterWeitenberg
Fluorescence Imaging of Quantum Gases
(a) 3
n=3
µ/U
2
1
0
11
T = 0 (b)
n=1
n=2
superfluid
y
n=2
x
(c)
n=1
Mott insulator
(J/U)c
J/U
Fig. 7.4. (a) Schematic zero-temperature phase diagram of the Bose Hubbard model as
a function of J/U and µ/U . The blue regions mark the Mott insulating state, the bright
grey region the superfluid. The transition to the n = 1 Mott lobe occurs at the critical
ratio (J/U )c , which depends on the dimensionality of the system ((J/U )1D
≈ 0.3 for
c
1D systems31,32 and (J/U )2D
≈ 0.06 for 2D systems33 ). The arrows indicate possible
c
measurements: the vertical arrows correspond to a measurement of the density profile in
a trapped system; the horizontal arrows correspond to measurements at fixed positions in
the trap, but at varying lattice depth, i.e. probing different values of J/U . (b) Schematic
of the shell structure of a Mott insulator in a trapped system with arrows indicating the
identification with the phase diagram. (c) Experimental profile of a Mott insulator with
four shells (with kind permission from Markus Greiner).
(c)
2
σdet (atoms2/site2) ndet (atoms/site)
(a)
1
0.5
0
0.25
(b)
0
0
2
4
6
Radial distance r (µm)
8
Fig. 7.5. (a),(b) High-resolution fluorescence images of Mott insulators for two different
particle numbers in the zero-tunneling limit (left panels) together with the numerically
reconstructed atom distributions on the lattice (right panels). Each circle indicates a
single atom; the points mark the lattice sites. (c) Radial atom density and variance
profiles obtained from an azimuthal average (corrected for the slight ellipticity of the
confining potential) of the reconstructed images in (a) and (b) (yellow and red points,
respectively). Each data set corresponds to a single image. The displayed statistical
error bars are Clopper Pearson 68 % confidence intervals for the binominally distributed
number of excitations. The solid lines show the result of a simultaneous fit to the
density and variance profiles
p with the model function of Eq. (7.1). The fit parameters
are T /U , µ/U and r0 = 2U/(mωx ωy ). The fits yield temperatures T = 0.090(5)U/kB
and T = 0.074(5)U/kB , chemical potentials µ = 0.73(3)U and µ = 1.17(1)U , and radii
r0 = 5.7(1) µm and r0 = 5.95(4) µm, respectively. The grey points were obtained from
a superfluid for reference. Figure adapted from Ref. 8.
0:15
World Scientific Review Volume - 9in x 6in
12
ChapterWeitenberg
Christof Weitenberg
(a) 6Er
(b) 16 Er
680nm
6
(c) 1.0
4
podd
Counts (x 1000)
2
0.00
0.10
n=1 shell
0.8
0.20
0.6
T/U = 0.05
T/U = 0.15
T/U = 0.20
0.4
0.2
0.0
n=2 shell
0
20
U/J
40
0.25
Variance
February 16, 2014
0.20
0.10
0.00
Fig. 7.6. High-resolution fluorescence images of atom number fluctuations across the
superfluid-to-Mott-insulator transition. (a)(b) Fluorescence images from a region of
10 × 8 lattice sites within the n = 1 Mott shell that forms in a deep lattice (upper
panel) together with the result from the detection algorithm (lower panel). Solid and
open circles mark the presence or absence of an atom, respectively. (a) In the superfluid
regime (lattice depth 6Er ), sites can be occupied with odd or even atom numbers, which
appear as full or empty sites, respectively. (b) In the Mott insulator (lattice depth 16Er ),
occupancies other than 1 are highly suppressed. (c) Measured value of the probability
podd for an odd occupation vs. the interaction-to-tunneling ratio U/J. Data sets, with
1σ error bars, are shown for regions that form a part of the n = 1 (squares) and n = 2
(circles) Mott shells in a deep lattice. The lines are based on finite-temperature Monte
Carlo simulations in a homogeneous system at constant temperature-to-interaction ratio
T /U of 0.20 (dotted red line), 0.15 (solid black line) and 0.05. The average temperature
extracted using the data points at the three highest U/J ratios is T /U ≈ 0.16 ± 0.03.
Figure adapted from: W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S.
Fölling, L. Pollet, and M. Greiner, Probing the superfluid-to-Mott insulator transition
at the single-atom level, Science 329, 547–550, (2010). Reprinted with permission from
AAAS.
in the zero-tunneling limit. It is a valuable tool, since thermometry in a
lattice is notoriously difficult.25
The on-site number fluctuations are an important indicator for the physical state. Let us consider how the number fluctuations are modified by the
parity projection. As the detected occupations ndet are either zero or one,
squaring them has no effect, i.e. n2det = ndet and also n¯2 det = n̄det . There2
fore the variance σdet
is directly connected to the mean occupation via
2
σdet = n̄det (1 − n̄det ). The mean and the variance of the measured occupation are not independent. In a superfluid, the atoms are delocalized
over the lattice and the initial occupations after pinning follow a Poisson
distribution. For a sufficiently high mean density, one has an even or odd
occupation with equal probability, and after parity projection, the mean
detected occupation is n̄det = 0.5 corresponding to the maximal variance
2
σdet
= 0.25. In a Mott insulator, the density is pinned to an integer number
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
Fluorescence Imaging of Quantum Gases
ChapterWeitenberg
13
of atoms per sites and the fluctuations vanish. The parity projection will
bring this to a mean occupation of n̄det = 1 or 0 for odd or even numbered
Mott shells, respectively.
Figure 7.6 demonstrates the reduction of the number fluctuations across
the transition from the superfluid to the Mott insulator state.7 In these
measurements, two regions of 10 × 8 lattice sites were chosen, at positions
where a n = 1 Mott shell and a n = 2 Mott shell form in a deep lattice,
respectively and the mean occupationf was obtained by averaging over the
small region and over repeated runs of the experiment. The data shows
how it goes smoothly from ∼ 0.5 to close to 1 or 0 in the n = 1 shell
and the n = 2 shell, respectively. Note also the right axis indicating how
the variance of the occupation number goes from ∼ 0.25 to close to 0. As
expected from the phase diagram in Fig. 7.4, the n = 1 shell reaches the
full number squeezing already for smaller values of U/J than the n = 2
shell (note that the x-axis is inverted in this figure). Residual number
fluctuations, which limit the mean occupation to n̄det = 94.9% ± 0.7% in
the n = 1 shell even deep in the Mott insulator, are thermal fluctuations
and can be used to deduce the temperature.
7.3.2. Parity parity correlations
In situ fluorescence imaging can use its full potential when it comes to
density density correlations. Often the nearest-neighbor correlations are
strongest and therefore a true single-site resolution is required. Here we
want to discuss the parity parity correlations in 1D systems across the
superfluid Mott insulator transition,15 revealing the strong correlations
around the transition.
In a Mott insulator at zero temperature and in the zero-tunneling limit,
atom number fluctuations are completely suppressed. If one adds a finite tunnel coupling, one introduces quantum fluctuations in the form of
correlated particle-hole pairs on top of this fixed-density background [see
Fig. 7.7(a)], which appear with a fraction proportional to (J/U )2 within
first-order perturbation theory. Closer to the transition to the superfluid
the particle-hole pairs grow in size and eventually unbind at the transition
point.
We consider in the regime of a Mott insulator of mean density n̄ = 1 and
f The
measured mean occupation n̄det can be interpreted as the probability podd that
the occupation number before the parity projection was odd. (Because the atoms are
lost pairwise, sites with initially even occupations will appear empty). This naming was
chosen by the Harvard group7 and is used here in Fig. 7.6
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
14
ChapterWeitenberg
Christof Weitenberg
(a)
(c)
0.2
MI
C(d=1)
0.15
(b)
SF
0.1
y
0.05
0
x
0
0.1
J/U
0.2
0.3
Fig. 7.7. Nearest-neighbor correlations in 1D. (a) Schematic image of the density distribution for 0 < J/U (J/U )c with parallel 1D tubes and particle-hole pairs aligned in
the direction of the tubes. (b) Typical experimental fluorescence images for J/U = 0.06
(left panel) together with the numerically reconstructed atom distribution (right panel).
Particle-hole pairs are emphasized by a yellow shading. The correlations C(d) are obtained by an average over many experimental realizations and by an additional average
over j in a central region of 9 × 7 sites indicated by the red box. The 1D tubes are
created by keeping the lattice axis along y at a constant depth of Vy = 17(1)Er , while
the lattice axis along the x-axis was varied to probe different values of J/U . (c) 1D
nearest-neighbor correlations Cp (d = 1) as a function of J/U . The data along the tube
direction (x-direction, red circles) shows a positive correlation signal, while the signal in
the orthogonal direction (y-direction, blue circles) vanishes within error bars due to the
decoupling of the 1D systems. The curves are first-order perturbation theory (dasheddotted line), density matrix renormalization group calculations for a homogeneous system at temperature T = 0 (dashed line) and finite-temperature matrix product state
calculations including harmonic confinement at T = 0.09U/kB (solid line). Thermal
fluctuations do not contribute to the correlation signal, but instead lead to a reduction
of the signal, as can be seen from the comparison of the theory curves for finite and
zero temperature. The error bars denote the 1σ statistical uncertainty. The light blue
shading highlights the superfluid phase. Figure adapted from Ref. 15.
define the deviation δn̂j = n̂j − n̄ in the occupation of the jth lattice site
(with the occupation operator n̂j ) from the average background density.
We define the parity operator ŝj = eiπδn̂j at site j. Now ŝj yields +1 for an
odd occupation number nj and −1 for an even nj . We consider the two-site
parity correlation functions at a distance d
C(d) = hŝj ŝj+d i − hŝj i hŝj+d i
(7.2)
and we will limit ourselves to nearest-neighbor correlations d = 1 here.
If there is a particle-hole pair on sites j and j + d, the same parity
s(nj ) = s(nj + d) = 1 is detected on these sites. Therefore hŝj ŝj+d i obtains a larger value than the factorized form hŝj i hŝj+d i, yielding a signal
in C(d). The factorized form captures uncorrelated fluctuations, for exam-
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
Fluorescence Imaging of Quantum Gases
ChapterWeitenberg
15
ple thermally activated particles or holes. Defects in the Mott insulator at
finite J/U and finite T can stem from thermal or quantum fluctuations.
Measuring correlation functions allows distinguishing of the quantum fluctuations, which leads to correlated particle-hole pairs and a finite C(d) from
thermally activated excitations, which are uncorrelated and yield C(d) = 0.
The experimental results for the nearest-neighbor correlations C(d = 1)
as a function of J/U are shown in Fig. 7.7(c). Deep in the Mott insulator, only uncorrelated thermal excitations exist and C(d) vanishes. With
increasing J/U , particle-hole pairs emerge and the nearest-neighbor correlations increase. They reach a peak value well before the transition to the
superfluid state. For even larger J/U , the correlations decrease and vanish in the non-interacting limit U = 0. The behavior is well captured by
a finite-temperature matrix product state calculation including harmonic
confinement.
The measurements were repeated for 2D Mott insulators yielding a similar curve, but here the maximal correlation is reached at the transition
point. Furthermore, the correlations are weaker in the 2D system, because
quantum fluctuations play a less important role in higher dimensions. A
more detailed account of the measurement of correlation functions can be
found in Ref. 29.
7.4. Single-site addressing
7.4.1. Manipulation of individual spins
With a high-resolution imaging system implemented, it is possible to
project arbitrary light patterns onto the atoms with structures on the scale
of a lattice spacing. This can be useful to create sharp walls as boundary
conditions or to add controlled disorder. The simplest case is to send a collimated laser beam backwards through the imaging objective and to focus
it down on a single lattice site [Fig. 7.8(a)]. In the MPQ experiment, the
beam had a full-width-at-half-maximum (FWHM) of 600 nm.g Its position
was controlled by changing the angle before the objective with a two-axis
piezo mirror, and it could be positioned on a given lattice site with a precision of better than 0.1alat using an independent calibration measurement of
its position together with a feedback that tracks the slowly varying lattice
position.
g The
beam was superimposed with the imaging path via the reflection of an uncoated
glass window.
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
16
ChapterWeitenberg
Christof Weitenberg
(a)
b
Addressing Laser Beam
Microwave
Horn
6.8 GHz
(b)
1
y
x
alat = 532 nm
Atoms in 2D Optical Lattice
(c)
Energy / h (a.u.)
ω0
∆LS
σMW
ωMW
0
0
-0.5
y
x
2 µm
0
0.5
x Position (µm)
1
Fig. 7.8. Single-site addressing in a Mott insulator: (a) Atoms are prepared in a Mott
insulator with unity filling in spin state |0i. An off-resonant addressing laser beam is
focused down to a size of 600 nm (FWHM) and positioned on selected lattice sites with
an accuracy of better than 0.1alat . (b) Energy diagram of atoms in the lattice for the two
hyperfine states |0i and |1i (energy separation ~ω0 ). The addressing beam induces an
additional light shift on the atoms, following its intensity profile indicated by the dashed
line. The addressing beam is σ − polarized and has the magic wavelength 787.55 nm, such
that the light shift on state |0i vanishes. The light shift on state |1i leads to a detuning
∆LS = 2π(−70 kHz) of the transition between the two states, making the atom spectrally
addressable by the microwave field. A Landau Zener sweep (central frequency ωMW ,
sweep width σMW = 2π(60 kHz), 20 ms duration) transfers the addressed atoms from |0i
to |1i. The lattice depths during the addressing were set to Vx = 56Er , Vy = 90Er and
Vz = 70Er in order to completely suppress tunneling even when the addressing beam
locally perturbs the lattice potential. The addressing beam is ramped up and down
within 2.5 ms in order to prevent vibrational excitations. The intensity of the addressing
beam is a compromise for yielding a sufficiently large differential light shift that can be
resolved in the presence of magnetic field fluctuations without leading to too strong a
deformation of the lattice potential. A magnetic bias field in the z-direction is used to
provide a quantization axis and creates a shift of −570 kHz on the bare resonance. (c)
Fluorescence images of the created spin pattern. The spins were flipped from |0i to |1i
in a line of 16 atoms. Left panel: image after the atoms in state |1i were removed by a
resonant laser pulse. Right panel: a global microwave sweep exchanged the populations
in |0i and |1i, such that only the addressed atoms are observed. (The two images stem
from two independent preparations of the system). Figure adapted from Ref. 19.
The beam was used to manipulate the spin of individual atoms.19 Two
hyperfine levels |0i = |F = 1, mF = −1i and |1i = |F = 2, mF = −2i, connected by a microwave transition at 6.8 GHz, were identified as pseudo-spin
states, and the atoms were initially prepared in a n = 1 Mott insulator in the
|0i state. The addressing beam created a differential light shift between the
spin states a the selected atom, thus making it spectrally addressable with
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
Fluorescence Imaging of Quantum Gases
ChapterWeitenberg
17
the microwave field [see Fig. 7.8(b)]. A Landau Zener sweep off-resonant
with the other atoms was then used to flip the selected spin to the |1i state.
The combination of a differential light shift and microwave pulses allows
reaching sub-diffraction-limited resolution. Although the addressing beam
intensity at the neighboring site is still 10 % of the maximum intensity, their
spin-flip probability is non-discernable from the background.
The scheme was used to create arbitrary spin pattern in the Mott insulator by repeating the procedure on several lattice sites.h The fluorescence
imaging is not spin-sensitive, because the optical molasses mixes all hyperfine levels. However, one can image the spin indirectly, by selectively
removing one of the spin states with a resonant push-out pulse before the
imaging (a pulse of 5 ms duration resonant with the F = 2 to F 0 = 3
transition efficiently removes the atoms from the |1i state without affecting
the atoms in |0i). An identification of the empty sites with the removed
spin state is possible if one can ensure that there was exactly one atom
before the push out. The low defect density in the n = 1 Mott insulators
of Fig. 7.5(a) show that this can indeed be realized with high fidelity.
Figure 7.8(c) shows fluorescence images of an n = 1 Mott insulator in
state |0i, in which the spins were flipped to |1i along a line of 16 atoms.
In the left panel the state |0i was removed. In the right panel, a global
microwave sweep exchanged the population in |0i and |1i before the push
out, such that only the addressed atoms were observed. In these first experiments, the spin flip fidelity was 95 %.
7.4.2. Coherent tunneling dynamics
The preparation of an arbitrary atom distribution opens up new possibilities
for exploring coherent quantum dynamics at the single-atom level. As an
example we want to discuss the tunneling dynamics in a 1D lattice.19 In
the experiment a single line of up to 18 atoms was prepared along the ydirection [Fig. 7.9(a)]. Then the lattice along the x-direction was lowered to
Vx = 5.0(5)Er within 200 µs to start the dynamics. (The lattice along y was
kept at 30 Er to keep the 1D tubes decoupled). After a varying evolution
time t, the atomic distribution was frozen by a rapid 100 µs ramp of all
lattice axes to more than 50 Er . The probability distribution of finding the
atom at different lattice sites was obtained by averaging the resulting atom
h In
an update of the experiment, the target sites were addressed simultaneously with
a light pattern created by a digital mirror device and the spins were flipped with a
single microwave pulse.20 This multiple-site addressing technique allows for a faster
preparation of the spin pattern leading to less heating.
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
18
ChapterWeitenberg
Christof Weitenberg
(a)
(b)
(c)
(d)
y
Probability
x
1
0.0 ms
0.5
0
−10
−5
0
5
x Position (alat)
0.4
0.3
0.2
0.1
0
10 −10
0.75 ms
−5
0
5
x Position (alat)
0.4
0.3
0.2
0.1
0
10 −10
1.5 ms
−5
0
5
x Position (alat)
0.4
0.3
0.2
0.1
0
10 −10
3.0 ms
−5
0
5
x Position (alat)
10
Fig. 7.9. Coherent tunneling dynamics. A single line of atoms along the y-direction was
prepared using the addressing scheme (white circles indicate the lattice sites at which
the atoms were prepared. Not all sites initially contained an atom). The lattice along
the x-direction was lowered to Vx = 5.0(5)Er for a varying evolution time, allowing the
atoms to tunnel in this direction. (a) to (d) Top row: fluorescence images of the atomic
distribution after different evolution times. Bottom row: position distributions obtained
from an average over the 1D tubes within each picture and over 10 to 20 such pictures.
The error bars give the 1σ statistical uncertainty. A fit to all distributions recorded at
different hold times (solid lines) yields a tunneling coupling of J/h = 940(20) Hz, a trap
frequency of ωx /(2π) = 103(4) Hz and a trap offset of xoff = −6.3 alat . Figure adapted
from Ref. 19.
positions over the 1D tubes and repeating the experiment several times.
This distribution samples the single-atom wavefunction as it expands in
the lattice. The interference of different path leads to distinct maxima and
minima in the distribution (Fig. 7.9).
The tunneling dynamics can be described by a simple Hamiltonian including the tunnel coupling J and the external confinement of trap frequency ωx with a position offset xoff from the prepared initial position,19
which is responsible for the asymmetry of the distribution for larger evolution times. A single fit to all probability distributions recorded at different
evolution times yields a tunnel coupling J/h = 940(20) Hz in good agreement with the independently measured lattice depth.
The same technique was applied to study the dynamics of a spin impurity in a 1D lattice.20 A line of flipped spins was created in the Mott
insulator, and the 1D dynamics were observed at various lattice depths. The
interaction with the other spin component strongly modified the dynamics.
In the Mott insulating regime, the impurity dynamics show the same kind
of distribution, but with a slower dynamics given by the superexchange
coupling Jex = 4J 2 /U . In the superfluid regime the spin impurity couples
to a bath that has quantum fluctuations and the description is more in-
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
Fluorescence Imaging of Quantum Gases
ChapterWeitenberg
19
volved. Comparing the width of the distributions one finds an expansion
velocity that is roughly half that of the free particle discussed above.
7.5. Perspectives
In this chapter, we have described the technique of single-atom resolved
fluorescence imaging. We discussed its technical implementation and presented its application to Bose Hubbard physics. The method offers a wide
range of new possibilities in quantum gas experiments. It could be used to
detect new observables such as entanglement in the many-body system35,36
or the Greens function via many-body Ramsey interferometry.37 An exciting advance would be to adapt the technique to fermionic species in optical
lattices in order to directly detect antiferromagnetic order38 or the FFLO
state39 at low temperature.
The local addressing capabilities open the path to novel cooling schemes
that rely on the local removal of regions with high entropy.13,40 Cold atoms
in optical lattices are also a promising candidate for scalable quantum
computing. The almost defect-free Mott insulators of several hundreds of
atoms that were demonstrated would form a well initialized quantum register. The single-spin control via the addressing beam could be extended
to single-qubit gates and the two-qubit gates could be realized via Rydberg
blockade.41,42
References
1. W. Neuhauser, M. Hohenstatt, P. E. Toschek, and H. Dehmelt, Localized
visible Ba+ mono-ion oscillator, Phys. Rev. A. 22, 1137–1140, (1980).
2. N. Schlosser, G. Reymond, I. Protsenko, and P. Grangier, Sub-Poissonian
loading of single atoms in a microscopic dipole trap, Nature. 411, 1024–1027,
(2001).
3. D. Schrader, I. Dotsenko, M. Khudaverdyan, Y. Miroshnychenko,
A. Rauschenbeutel, and D. Meschede, Neutral atom quantum register, Phys.
Rev. Lett. 93, 150501, (2004).
4. K. D. Nelson, X. Li, and D. S. Weiss, Imaging single atoms in a threedimensional array, Nature Physics. 3, 556–560, (2007).
5. T. Gericke, P. Würtz, D. Reitz, T. Langen, and H. Ott, High-resolution
scanning electron microscopy of an ultracold quantum gas, Nature Physics.
4, 949–953, (2008).
6. W. S. Bakr, J. I. Gillen, A. Peng, S. Fölling, and M. Greiner, A quantum gas
microscope for detecting single atoms in a Hubbard-regime optical lattice,
Nature. 462, 74–77, (2009).
February 16, 2014
20
0:15
World Scientific Review Volume - 9in x 6in
ChapterWeitenberg
Christof Weitenberg
7. W. S. Bakr, A. Peng, M. E. Tai, R. Ma, J. Simon, J. I. Gillen, S. Fölling,
L. Pollet, and M. Greiner, Probing the superfluid-to-Mott insulator transition
at the single-atom level., Science. 329, 547–550, (2010).
8. J. F. Sherson, C. Weitenberg, M. Endres, M. Cheneau, I. Bloch, and S. Kuhr,
Single-atom-resolved fluorescence imaging of an atomic Mott insulator, Nature. 467, 68–72, (2010).
9. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, and P. Zoller, Cold bosonic
atoms in optical lattices, Phys. Rev. Lett. 81, 3108–3111, (1998). doi: 10.
1103/PhysRevLett.81.3108.
10. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, and I. Bloch, Quantum
phase transition from a superfluid to a Mott insulator in a gas of ultracold
atoms, Nature. 415, 39–44, (2002).
11. N. Gemelke, X. Zhang, C.-L. Hung, and C. Chin, In situ observation of
incompressible Mott-insulating domains in ultracold atomic gases, Nature.
460, 995–998, (2009).
12. M. Endres, T. Fukuhara, D. Pekker, M. Cheneau, P. Schauß, C. Gross,
E. Demler, S. Kuhr, and I. Bloch, The ‘Higgs’ amplitude mode at the
two-dimensional superfluid/Mott insulator transition, Nature. 487, 454–458,
(2012).
13. W. S. Bakr, P. M. Preiss, M. E. Tai, R. Ma, J. Simon, and M. Greiner, Orbital
excitation blockade and algorithmic cooling in quantum gases, Nature. 480
(7378), 500–503, (2011).
14. R. Ma, M. E. Tai, P. M. Preiss, W. S. Bakr, J. Simon, and M. Greiner,
Photon-assisted tunneling in a biased strongly correlated Bose gas, Phys.
Rev. Lett. 107, 095301, (2011).
15. M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauß, C. Gross,
L. Mazza, M. C. Banuls, L. Pollet, I. Bloch, and S. Kuhr, Observation of
correlated particle-hole pairs and string order in low-dimensional Mott insulators, Science. 334, 200–203, (2011).
16. M. Cheneau, P. Barmettler, D. Poletti, M. Endres, P. Schauß, T. Fukuhara,
C. Gross, I. Bloch, C. Kollath, and S. Kuhr, Light-cone-like spreading of
correlations in a quantum many-body system, Nature. 481, 484–487, (2012).
17. P. Schauß, M. Cheneau, M. Endres, T. Fukuhara, S. Hild, A. Omran, T. Pohl,
C. Gross, S. Kuhr, and I. Bloch, Observation of spatially ordered structures
in a two-dimensional Rydberg gas, Nature. 491, 87–91, (2012).
18. B. Zimmermann, T. Müller, J. Meineke, T. Esslinger, and H. Moritz, Highresolution imaging of ultracold fermions in microscopically tailored optical
potentials, New J. Phys. 13, 043007, (2011).
19. C. Weitenberg, M. Endres, J. F. Sherson, M. Cheneau, P. Schauß,
T. Fukuhara, I. Bloch, and S. Kuhr, Single-spin addressing in an atomic
Mott insulator, Nature. 471, 319–324, (2011).
20. T. Fukuhara, A. Kantian, M. Endres, M. Cheneau, P. Schauß, S. Hild,
D. Bellem, U. Schollwöck, T. Giamarchi, C. Gross, I. Bloch, and S. Kuhr,
Quantum dynamics of a mobile spin impurity, Nature Physics. 9, 235–241,
(2013).
21. T. Fukuhara, P. Schauß, M. Endres, S. Hild, M. Cheneau, I. Bloch, and
February 16, 2014
0:15
World Scientific Review Volume - 9in x 6in
Fluorescence Imaging of Quantum Gases
22.
23.
24.
25.
26.
27.
28.
29.
30.
31.
32.
33.
34.
35.
36.
37.
ChapterWeitenberg
21
C. Gross, Microscopic observation of magnon bound states and their dynamics, Nature. 502, 76–79, (2013).
L.-M. Duan, E. Demler, and M. Lukin, Controlling spin exchange interactions
of ultracold atoms in optical lattices, Phys. Rev. Lett. 91, 090402, (2003).
S. Sachdev, K. Sengupta, and S. M. Girvin, Mott insulators in strong electric
fields, Phys. Rev. B. 66, 075128, (2002).
J. Simon, W. S. Bakr, R. Ma, M. E. Tai, P. M. Preiss, and M. Greiner,
Quantum simulation of antiferromagnetic spin chains in an optical lattice,
Nature. 472, 307–312, (2011).
D. C. McKay, D. Jervis, D. J. Fine, J. W. Simpson-Porco, G. J. A. Edge, and
J. H. Thywissen, Low-temperature high-density magneto-optical trapping of
potassium using the open 4S → 5P transition at 405 nm, Phys. Rev. A. 84,
063420, (2011).
P. M. Duarte, R. A. Hart, J. M. Hitchcock, T. A. Corcovilos, T.-L. Yang,
A. Reed, and R. G. Hulet, All-optical production of a lithium quantum gas
using narrow-line laser cooling, Phys. Rev. A. 84, 061406, (2011).
M. T. DePue, C. McCormick, S. L. Winoto, S. Oliver, and D. S. Weiss, Unity
occupation of sites in a 3D optical lattice, Phys. Rev. Lett. 82, 2262–2265,
(1999).
A. Fuhrmanek, R. Bourgain, Y. R. P. Sortais, and A. Browaeys, Study of
light-assisted collisions between a few cold atoms in a microscopic dipole trap,
Phys. Rev. A. 85, 062708, (2012).
M. Endres, M. Cheneau, T. Fukuhara, C. Weitenberg, P. Schauß, C. Gross,
L. Mazza, M. C. Bañuls, L. Pollet, I. Bloch, and S. Kuhr, Single-site- and
single-atom-resolved measurement of correlation functions, Appl. Phys. B.
113, 27–39, (2013).
M. Karski, L. Förster, J. Choi, W. Alt, A. Widera, and D. Meschede, NearestNeighbor detection of atoms in a 1D optical lattice by fluorescence imaging,
Phys. Rev. Lett. 102, 053001, (2009).
T. D. Kühner, S. R. White, and H. Monien, One-dimensional Bose Hubbard
model with nearest-neighbor interaction, Phys. Rev. B. 61, 12474, (2000).
V. Kashurnikov, A. Krasavin, and B. Svistunov, Mott-insulator–superfluidliquid transition in a one-dimensional bosonic Hubbard model: Quantum
Monte Carlo method, JETP Lett. 64, 99–104, (1996).
B. Capogrosso-Sansone, S. G. Söyler, N. Prokofev, and B. Svistunov, Monte
Carlo study of the two-dimensional Bose Hubbard model, Phys. Rev. A. 77,
015602, (2008).
F. Gerbier, Boson Mott insulators at finite temperatures, Phys. Rev. Lett.
99, 120405, (2007).
A. J. Daley, H. Pichler, J. Schachenmayer, and P. Zoller, Measuring entanglement growth in quench dynamics of bosons in an optical lattice, Phys.
Rev. Lett. 109, 020505, (2012).
H. Pichler, L. Bonnes, A. J. Daley, A. M. Läuchli, and P. Zoller, Thermal
versus entanglement entropy: A measurement protocol for fermionic atoms
with a quantum gas microscope, New J. Phys. 15, 063003, (2013).
M. Knap, A. Kantian, T. Giamarchi, I. Bloch, M. D. Lukin, and E. Demler,
February 16, 2014
22
38.
39.
40.
41.
42.
0:15
World Scientific Review Volume - 9in x 6in
ChapterWeitenberg
Christof Weitenberg
Probing real-space and time-resolved correlation functions with many-body
Ramsey interferometry, Phys. Rev. Lett. 111, 147205, (2013).
D. Greif, T. Uehlinger, G. Jotzu, L. Tarruell, and T. Esslinger, Short-range
quantum magnetism of ultracold fermions in an optical lattice, Science. 340,
1307–1310, (2013).
M. O. J. Heikkinen, D.-H. Kim, and P. Törmä, Finite-temperature stability
and dimensional crossover of exotic superfluidity in lattices, Phys. Rev. B.
87, 224513, (2013).
J.-S. Bernier, C. Kollath, A. Georges, L. De Leo, F. Gerbier, C. Salomon,
and M. Köhl, Cooling fermionic atoms in optical lattices by shaping the
confinement, Phys. Rev. A. 79, 061601(R), (2009).
T. Wilk, A. Gaetan, C. Evellin, J. Wolters, Y. Miroshnychenko, P. Grangier, and A. Browaeys, Entanglement of two individual neutral atoms using
Rydberg blockade, Phys. Rev. Lett. 104, 010502, (2010).
L. Isenhower, E. Urban, X. L. Zhang, A. T. Gill, T. Henage, T. A. Johnson,
T. G. Walker, and M. Saffman, Demonstration of a neutral atom controlledNOT quantum gate, Phys. Rev. Lett. 104, 010503, (2010).