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Quantum gas magnifier for sub-lattice-resolved imaging of 3D quantum systems | Nature
Methods . Optical lattice setup . Our optical lattice setup consists of three running waves of wave vector k i with k i ?=?2π/ λ intersecting under an angle of 120°. Depending on the polarization of the beams we obtain either a triangular lattice (linear polarization perpendicular to the lattice plane), a honeycomb lattice (linear polarization in plane) 37 or a boron nitride lattice (suitable elliptical polarization of the lattice beams 38 as in this work or using spin-dependent light shifts 39 ). The resulting potential can be written as \(\begin{array}{c}{V}_{{\rm{lat2D}}}({\bf{r}})=\sum _{i > j}\sqrt{{V}_{{\rm{lat}}}^{(i)}{V}_{{\rm{lat}}}^{(j)}}\\ \,\times [{\cos }^{2}(\theta )\,\cos (({{\bf{k}}}_{i}-{{\bf{k}}}_{j}){\bf{r}}+{\alpha }_{i}-{\alpha }_{j})-2{\sin }^{2}(\theta )\,\cos (({{\bf{k}}}_{i}-{{\bf{k}}}_{j}){\bf{r}})]\end{array}\) where the \({V}_{{\rm{lat}}}^{(i)}\) are proportional to the intensities of the lattice beams. θ is the angle of the polarization (long half axis) with respect to the lattice plane, α i is the relative phase between the s and p components of the polarization for beam i . We neglected the phases of the beams with respect to each other because they only result in a global shift of the lattice. If we just name a single lattice depth, then all \({V}_{{\rm{lat}}}^{(i)}\) are equal. The boron nitride lattice in Fig. 3 uses θ ?=?9° and α ?=?(0,120°,240°) yielding an energy offset between the A and B sublattice quantified by the tight-binding parameter Δ AB (ref. 38 ). Note that the triangular lattice has a much larger barrier between nearest neighbours than the honeycomb or boron nitride lattice for the same laser intensities 40 . Read-out of lattice site populations . For several experiments only the total population of the lattice sites is of interest. We extract these by first fitting a triangular lattice to the data and subsequently summing up the signal in the Wigner–Seitz cells around the individual sites as explained in the following. The lattice constant a lat in pixels is determined by integrating the density of individual images along a real space lattice vector yielding a one-dimensional profile with lattice constant a 1D , which is obtained from a fit with the heuristic function A exp(?( x ??? x 0 ) 2 /(2 σ 2 ))(cos(π x / a 1D ?+? ? ) 2 ?+? Δ ). Finally, the lattice constant is deduced from the average fit parameter from two different such directions as \({a}_{{\rm{lat}}}=2{a}_{1{\rm{D}}}/\sqrt{3}\) . Next, the spatial phase of the lattice is determined by multiplying the image with a mask that removes the signal from pixels at a certain radius around the sites of a triangular lattice with the lattice constant determined beforehand. The phase of this mask is varied and the configuration minimizing the remaining density is considered the lattice phase. The final step is to determine the population of each lattice site by summing over the Wigner–Seitz cell around the lattice site. To minimize discretization errors the pixels of the camera are subdivided such that the radius of the cell is about ten subpixels. For an example image with non-discretized Wigner–Seitz masks see Extended Data Fig. 1 . For the lattices with two-atomic basis we slightly adjust the algorithm for lattice phase determination by maximizing the density which is not masked thus locating the centres of the honeycombs. Lattice phase drifts . For our hexagonal lattice setup composed of three laser beams in two dimensions, phase shifts of the lattice beams only lead to a translation of the whole lattice potential, but not to a change of the lattice geometry 41 . We verify that such phase drifts are not a problem on the time scale of the experiments presented here by measuring the position drift of the atomic cloud’s centre of mass in a very deep optical lattice. We find that the cloud position moves and scatters by less than one lattice site peak-to-peak within 6?s hold time. We checked in a previous set of measurements where we deliberately move the lattice, that the lattice is deep enough to be able to drag the atoms along. Shot-to-shot lattice drifts exceed one lattice site (cycle time of 30?s). Our characterization of the slow phase drifts is compatible with recent direct measurements of triangular lattices using quantum gas microscopes 42 , 43 . The drifts can be further reduced to one lattice site per minute in a setup with a single, refolded lattice beam 43 . In our case, the three beams go through separate optical fibres, a setup in which phase locks have been implemented to stabilize the phase 37 . From our characterization, we conclude that a phase lock is not necessary for the measurements presented here. The random lattice phase between individual images can be easily taken into account by identifying the phase. For data evaluation in the main text, we determine the lattice position for every experimental image via a fit routine as described above. Note that the envelope of the atomic density is given by the position of the magnetic trap and is therefore not affected by lattice phase drifts. Bimodal fits of density profiles . The lattice-gas profiles can be described by a bimodal model. Since we are considering the on-site populations only, the presence of the lattice can be included by a renormalization of the interaction constant 44 g eff ?=? g ?×? A WS /(2π σ 2 ) and otherwise using a continuum formalism. Here, A WS is the area of the Wigner–Seitz cell, σ the on-site radial oscillator length and g ?=?4π ? 2 a sc / m the interaction constant, computed from the scattering length a sc ?≈?100 Bohr radii and the mass m ?=?87?u. The on-site radial oscillator length is computed as \(\sigma =\sqrt{\hbar /(m{\omega }_{{\rm{onsite}}})}\) from the lattice depth using \(\hbar {\omega }_{{\rm{onsite}}}=3\sqrt{2{V}_{{\rm{lat}}}/{E}_{{\rm{rec}}}}{E}_{{\rm{rec}}}\) . The data in Fig. 2 is taken with a lattice depth of V lat ?=?1 E rec . The condensed atoms are described by a 3D Thomas–Fermi profile integrated along line of sight, $${n}_{{\rm{BEC}}}(x,y)=\int {\rm{d}}{z}\frac{15}{8{\rm{\pi }}}\frac{{N}_{{\rm{BEC}}}}{{{R}}_{\rho }^{2}{{R}}_{{z}}}\left(1-\frac{\rho {(x,y)}^{2}}{{{R}}_{\rho }^{2}}-\frac{{{z}}^{2}}{{{R}}_{{z}}^{2}}\right).$$ (1) The fit parameters here are the centre of the cloud x 0 , y 0 resulting in ρ ( x , y ) 2 ?=?( x ??? x 0 ) 2 ?+?( y ??? y 0 ) 2 , the in-plane Thomas–Fermi radius R ρ from which the out-of-plane radius R z is deduced via a computed aspect ratio, and the number of atoms in the BEC N BEC . In fact, only for the lowest evaporation frequency, where the BEC is very distinct from the thermal part, N BEC and R ρ are fitted independently. For all other fits we compute the Thomas–Fermi radius from the number of condensed atoms using the expected scaling R ρ ?=? γN BEC 1/5 with γ determined as its mean value from the fits at lowest evaporation frequency. We obtain γ ?=?0.354??m, which agrees excellently with the expected value γ theo ?=?0.352??m obtained from 45 $${\gamma }_{{\rm{theo}}}={15}^{1/5}\sqrt{\frac{\hbar \bar{\omega }}{m{\omega }_{{\rm{system}}}^{2}}}{\left(\frac{{g}_{{\rm{eff}}}}{g}\frac{{a}_{{\rm{sc}}}}{\bar{a}}\right)}^{1/5},$$ (2) supporting the validity of the approximations made. Here ω system ?=?2π?×?305 Hz, \(\bar{\omega }\) ?=? ( ω system 2 ω z ) 1/3 , ω z ?=?2π?×?29?Hz and \(\bar{a}=\sqrt{\hbar /(m\bar{\omega })}\) . The thermal density distribution is described in a semi-ideal approach, that is, as an ideal gas in a potential V ( x )?=? V trap ( x )?+? V BEC ( x ) given by the external trap V trap ( x ) and the repulsion from the condensed atoms V BEC ( x )?=?2 g eff n BEC ( x ). In semi-classical approximation the ideal Bose gas density distribution is given by 45 $${n}_{{\rm{th}}}(x)={g}_{3/2}(\exp (-\beta (V(x)-\mu )))/{\lambda }_{{T}}^{3}$$ (3) with \({g}_{n}(x)=\sum _{i > 0}{x}^{i}/{i}^{n}\) and \({\lambda }_{{T}}=\hbar \sqrt{2{\rm{\pi }}/(m{k}_{{\rm{B}}}{T})}\) . Additionally, we allow for a small offset that we subtract when determining atom numbers. The fit is performed on the 2D density distribution and both the data and the fit function are subsequently plotted as a function of radial position. Extended Data Fig. 2 shows the data from Fig. 2c, d of the manuscript along with a plot of the logarithm of the density versus the square of the radius, which yields a straight line in the thermal wings. This plot shows the excellent agreement between data and fit and also makes the change of the slope at the onset of the BEC fraction more visible. Interaction shift and finite size shift . Interactions are known to shift the critical temperature for the BEC transition with a sign depending on the trapping geometry. For a 3D harmonic trap in mean field approximation the shift is negative and given by 20 , 45 $$\Delta {T}_{{\rm{c}}}/{T}_{{\rm{c}}}\approx -1.33\frac{{g}_{{\rm{eff}}}}{g}\frac{{a}_{{\rm{sc}}}}{\bar{a}}{N}^{1/6}$$ (4) predicting a shift of about ?0.24 for the typical atom number of the condensed samples of N ?=?5?×?10 4 , which is larger than the measured shift of ?0.099(4). However, for interactions of this strength the mean-field approximation overestimates the shift 22 . Note that we are not aware of a prediction for our 2D–3D crossover geometry of an array of tubes. Our measurements thus set a benchmark for future theoretical studies on the interesting setting of Josephson junction arrays. We also recall the prediction for the finite size shift of the critical temperature for a 3D harmonic trap. For an anisotropic harmonic trap with trap frequencies ω x , ω y , ω z and their geometric mean \(\bar{\omega }={({\omega }_{x}{\omega }_{y}{\omega }_{z})}^{1/3}\) and arithmetic mean ω m ?=?( ω x + ω y + ω z )/3, the shift is given by 20 , 45 . $$\Delta {T}_{{\rm{c}}}/{T}_{{\rm{c}}}\approx -0.73\frac{{\omega }_{{\rm{m}}}}{\bar{\omega }}{N}^{-1/3}.$$ (5) With our trapping frequencies of 2π?×?(305, 305, 29)?Hz, the anisotropy factor is \({\omega }_{{\rm{m}}}/\bar{\omega }=1.53\) and the expected shift is ?0.03 for our atom number of N ?≈?5?×?10 4 , that is, much smaller than observed. Note that both interactions and finite size effects can contribute to the shift. The observed smoothing over a range of almost 0.2 in rescaled temperature is only expected for much smaller atom numbers in the case of a 3D harmonic trap 46 . We therefore conclude that finite size effects are strongly enhanced in our 2D–3D crossover geometry of an array of tubes. We have verified that the small condensate fractions involved in the smoothened transition do not arise from fit artefacts of the bimodal profile to the density profiles. The good agreement with the curve for the visibility shown in Extended Data Fig. 4 is further evidence that the signal is physical and demands for further theoretical studies. Theoretical description of the density of states . We compare our data of the thermal-to-BEC phase transition to non-interacting calculations based on the density of states. To this end we compute the Hamiltonian matrix for our trap in position basis and diagonalize it. In the numerical spectrum we clearly recognize a crossover between two power laws as a slope change in the log–log plot of Extended Data Fig. 3a . The asymptotes of this crossover can be understood using analytical considerations. The high energy limit coincides with the well-known spectrum of a 3D harmonic trap resulting in $$N(E)=\frac{1}{6}{\left(\frac{E}{\hbar \bar{\omega }}\right)}^{3}$$ (6) states up to energy E . This is due to the fact that the gaps between higher bands are negligible compared to the band widths. So we have to count separately the first band states and harmonic oscillator-like states. For energies E ?external trap, which is Δ ?=?1/2 mω syst 2 a l at 2 ?=? h ?×?200?Hz for a site in the centre compared to a nearest neighbour. Hence the spectrum is given by $${E}_{ijk}=1/2m{\omega }_{{\rm{syst}}}^{2}{r}_{ij}^{2}+(k+1/2)\hbar {\omega }_{{\rm{z}}}$$ (7) with r ij being the distance of the lattice site indexed ij from the trap centre and k is the index for the z direction. A lengthy calculation leads to N ( E )?=?( E / E 0 ) 2 with \({E}_{0}=\sqrt{\hbar {A}_{{\rm{WS}}}m{\bar{\omega }}^{3}/{\rm{\pi }}}=h\times 57\) Hz. We can therefore find an approximation of the numerical result by the Ansatz $$N(E)={\left(\frac{E}{{E}_{0}}\right)}^{2}+\,{\rm{\max }}\left(\frac{1}{6}{\left(\frac{E-\hbar {{\Delta }}_{{\rm{g}}}}{\hbar \bar{\omega }}\right)}^{3},0\right)$$ (8) where Δ g is obtained from a simulation without external trap. The crossover between the two power laws appears here at the band gap Δ g , because the higher bandgaps are small and the lattice can be neglected at higher energies. This analytical model fits very well to the exact diagonalization up to the numerically accessible energies (Extended Data Fig. 3a ) while asymptotically reaching the known analytic limit of equation ( 6 ) for high energies. Now we turn to the detailed derivation of the theory curve for a non-interacting system (light-blue line in Fig. 2f ). From N ( E ) we obtain the density of states g ( E )?=?d N /d E , which in turn allows to numerically compute the critical temperature \({T}_{{\rm{c}}}^{0}(N)\) from $$N=\int {\rm{d}}Eg(E)/[\exp (E/({k}_{{\rm{B}}}{T}_{{\rm{c}}}^{0}))-1],$$ (9) that is, \({T}_{{\rm{c}}}^{0}(N)\) is the temperature yielding exactly N excited atoms for chemical potential μ ?=?0. The fraction of condensed atoms for a given temperature \(T < {T}_{{\rm{c}}}^{0}\) can be computed by first evaluating the number of excited atoms as $${N}_{{\rm{exc}}}=\int {\rm{d}}Eg(E)/[\exp (E/({k}_{{\rm{B}}}T))-1]$$ (10) and then inferring f 0 ?=?( N ??? N exc )/ N . Following these steps we can compute \({T}_{{\rm{c}}}^{0}\) and f 0 for every experimental data point from its measured particle number and temperature. The resulting theoretical values are plotted in Extended Data Fig. 3b . We find that these values can be approximated by \({f}_{0}=1-{(T/{T}_{{\rm{c}}}^{0})}^{\alpha }\) as obtained by assuming the density of states \(g(E)={C}_{\alpha }{E}^{\alpha -1}\) to be a power law 45 . Fitting the theoretical results for \({f}_{0}(T/{T}_{{\rm{c}}}^{0})\) with α as the fit parameter yields α ?=?2.69(1). The corresponding fit shown in Extended Data Fig. 3b is the same line as the light-blue line in Fig. 2f (Extended Data Fig. 3b ). Comparison to ToF data . For comparison, we also take momentum space images from ToF expansion at the same parameters and evaluate their visibility 47 , which is a measure of coherence in the system. We use circular masks around the Bragg peaks (Extended Data Fig. 4a ). The radius is determined by fitting the ToF data by a central bimodal distribution $$n({\bf{k}};\sigma ,{k}_{{\rm{R}}},{n}_{0,{\rm{G}}},{n}_{0,{\rm{P}}},{{\bf{k}}}_{0})={n}_{0,{\rm{G}}}\exp (-{({\bf{k}}-{{\bf{k}}}_{0})}^{2}/(2{\sigma }^{2}))+{n}_{0,{\rm{P}}}\,{\rm{\max }}(1-{({\bf{k}}-{{\bf{k}}}_{0})}^{2}/{k}_{{\rm{R}}}^{2},0)$$ (11) and a set of six inverse parabola n ( k ;? k R ,? n 0,P ,? k 0 )?=? n 0,P max(1???( k ??? k 0 )2/ k R 2 ,?0) spaced by a reciprocal lattice vector from the centre, resulting in the complete fit function reading $$\begin{array}{c}n({\bf{k}};\sigma ,{k}_{{\rm{R}},c},{k}_{{\rm{R}},{\rm{Bragg}}},{n}_{0,{\rm{G}}},{n}_{0,{\rm{P}},c},{n}_{0,{\rm{P}},{\rm{Bragg}}},{{\bf{k}}}_{0},{k}_{{\rm{reci}}})\\ \,=n({\bf{k}};\sigma ,{k}_{{\rm{R}},{\rm{c}}},{n}_{0,{\rm{G}}},{n}_{0,{\rm{P}},c},{{\bf{k}}}_{0})+\mathop{\sum }\limits_{j=1}^{6}n({\bf{k}};{k}_{{\rm{R}},{\rm{Bragg}}},{n}_{0,{\rm{P}},{\rm{Bragg}}},{{\bf{k}}}_{0}\\ \,+{k}_{{\rm{reci}}}(\cos \,j{\rm{\pi }}/3,\,\sin \,j{\rm{\pi }}/3))\end{array}$$ (12) where the variables separated by a semicolon are the fit parameters, the parameter k R,Bragg is used for the radius and the parameters k 0 ) and k reci for the position of the circular masks. We plot the visibility as a function of \(T/{T}_{{\rm{c}}}^{0}\) as obtained from the corresponding real space data (Extended Data Fig. 4b ). We plot the theory curve for the condensate fraction as a guide to the eye. This comparison shows that the real-space and momentum-space images give a compatible description of the system. The visibility and the condensate fraction vanish for the same temperatures (see Fig. 2f and Extended Data Fig. 4 ). This is in contrast to 3D optical lattices around unit filling, where a finite visibility also for the case of vanishing condensate fraction is observed 18 , 19 . In these experiments the critical temperatures are much smaller, of the order of a few tunnelling energies, and thus low-energy states that are not the ground state but still have short range phase-coherence are substantially populated yielding a finite visibility above the critical temperature. For our experimental temperatures of a few hundred tunnelling energies no other state than the ground state gets substantially populated. Details on magnetic resonance addressing . In order to engineer the density distributions shown in Fig. 3 , we used a trap frequency of ω addressing /2π?=?543?Hz for the first five images, of ω addressing /2π?=?658?Hz for the last image and different trap shifts and RF sequences. By shifting the magnetic trap perpendicularly to a real-space lattice vector by around 14??m, corresponding to approximately twice the system diameter, the curvature of the equipotential lines becomes negligible and the density patterns created byaddressing exhibit straight edges. In Fig. 3 the trap centre resonance frequency is ω c /2π?=?108?kHz for all images, except the last one of panel b where it is ω c /2π?=?67?kHz. The trap is shifted by 14.1??m for the first and third image, by 15.7??m for the second image and not shifted for the fourth to sixth image, but always shifted back to the position of the atoms before imaging. For the third image a constant RF pulse of 360?kHz is turned on for 200?ms. For the first image, an RF ramp from 360 to 290?kHz is used, leading to the depletion of all lattice sites from the centre of the cloud towards the centre of the shifted magnetic trap. Here, for the same RF ramp time (200?ms) we ramp over a wider range and therefore have to compensate the reduced time by which the resonance condition is met at each position by increasing the RF amplitude. In all protocols, Fourier broadening is negligible. Lattice phase fluctuations from shot to shot lead to one or two partially depleted rows in most images. The second image in Fig. 3 is created by applying two RF ramps. In this case the trap was shifted further to the side resulting in a higher energy difference to the target F ?=?2, m F ?=?1 state and thus we used ramps from 420 to 486?kHz and from 494 to 540?kHz with 200?ms each to target all sites except for the centre line. For the fourth to sixth image 100?ms were used as the RF duration. In the fourth image the outer wings of the distribution are cut via a RF ramp from 150 to 110?kHz. In the followingimages only a single frequency very close to the respective ω c , 108.5 and 67.2?kHz, is used to address a ring or a single lattice site. The third and fifth image also visualize the second difference between addressing with and without shifting the magnetic trap: the slope grows linearly from the centre, which leads to sharper resonances for shifted systems. Modelling of thermal hopping . The Arrhenius law is often used to describe chemical reaction rates, but also to model thermal hopping of continuously laser-cooled atoms in very deep optical lattices 48 . Here we use it to model the thermal hopping of ultracold atoms in our two-dimensional lattice. In contrast to quantum mechanical tunnelling through the barrier between two lattice sites, thermal hopping refers to motion that is activated thermally when the thermal energy allows to overcome the barrier. To good approximation, the activation energy for a hopping event can be identified with the potential barrier in the lattice, which is V B ?=?9 V lat in our triangular lattice convention. The Arrhenius law describes the hopping rate Γ h as the product of an attempt rate Γ a and the probability P ( E ?>? V B ) to sample an energy E above the barrier V B in the thermal distribution. The hopping rate can then be written as $${{\Gamma }}_{{\rm{h}}}\approx {{\Gamma }}_{{\rm{a}}}P(E > {V}_{{\rm{B}}})={{\Gamma }}_{{\rm{a}}}({\int }_{{V}_{{\rm{B}}}}^{{\rm{\infty }}}\exp (-E/{k}_{{\rm{B}}}T){\rm{d}}E)/({k}_{{\rm{B}}}T).$$ (13) To include quantum tunnelling, we add an offset rate, resulting in $${{\Gamma }}_{{\rm{h}}}\approx {{\Gamma }}_{{\rm{a}}}\,\exp (-{V}_{{\rm{B}}}/{k}_{{\rm{B}}}T)+{{\Gamma }}_{0}.$$ (14) In Fig. 3 , we model the temperature-dependent thermalization rate by the modified Arrhenius law of equation ( 14 ) and extract an activation barrier of V B ?=? k B ?×?2.4(6)?μK and an attempt rate of Γ a ?=?52(44)?Hz as well as an offset rate of Γ 0 ?=?0.23(8)?Hz, which we attribute to quantum tunnelling in higher bands. The barrier height for the calibrated lattice depth of V lat ?=?3 E r is V B / k B ?=?2.6?μK. We note that in contrast to quantum tunnelling, for thermal hopping the atoms can move over long distances in single hopping events. This enables the large-scale mass transport in Fig. 3 within few hopping events. Modelling of nanoscale dynamics . We describe here the numerical simulations shown in Fig. 4c . The simulations start with the ground state of the periodic potential with initial optical lattice beam intensities I 2 , I 3 ?=? I 1 . At time t ?=?0, I 2 and I 3 are set to 0.5 I 1 ; the intensities change on the intensity lock time scale of about 20??s. For every time step (5??s) we diagonalize the Hamiltonian in plane-wave basis of the instantaneous periodic potential and let the state evolve according to the instantaneous eigenstates and eigenvalues. Because the dimers are decoupled from each other, the bands are completely flat and all quasi-momenta are equivalent and we perform the calculations at the Γ point in the Brillouin zone. After the quench, 99.5% of the probability distribution of the time-evolved state is found to lie in the lowest six bands, demonstrating that the dynamics features interference between the two s bands and four p bands, the latter being the smallest in-plane excitations within a lattice site. The extracted atomic distribution in a cut of 65?nm width is plotted in Fig. 4c (left). In Fig. 4c (middle) the distribution is convoluted with a Gauss filter of 76?nm width, and summed with an offset, for comparison with the experimental data in Fig. 4c (right). The lattice depth used in the theory (32 E rec ; note that the tunnel barriers are much smaller in a honeycomb lattice compared to a triangular lattice of the same total depth) is calibrated from the comparison with the experiment. The external trap is not included in the analysis, because experimentally we don’t see any dependence of the dynamics on the position of the dimer with respect to the trap centre. .
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