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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. 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