Main

The interplay between disorder and interaction is central to the richness of condensed-matter physics as any real-life material will inevitably contain a certain degree of impurities and defects, and interparticle interactions are almost always present. While disorder tends to localize non-interacting particles, leading to Anderson localization1, interactions can counteract this, resulting in conducting ergodic states. More generally, the combination of disorder and interactions gives rise to rich physics governed by reduced or absent relaxation and transport, such as glassy dynamics or non-ergodic many-body localized systems, and forms one of the central topics in quantum statistical physics during the past decade2.

In bosonic systems, a hallmark of this interplay is the emergence of an additional ground-state phase called Bose glass. The Bose glass is an insulating but compressible phase without long-range phase coherence3,4. It was originally discussed purely as a ground state at zero temperature, but has been shown to extend to finite energy9,10,11,12. In the weakly interacting regime, the Bose glass can be understood by starting from a non-interacting Anderson insulator, where in the ground state all bosons localize at the lowest potential minimum (Fig. 1c). Adding small repulsive interactions to such systems will lead to bosons spilling over into other low-lying orbitals to minimize the interaction energy. This regime has also been referred to as an Anderson glass or Lifshitz glass13,14. With increasing interactions or density, and thereby increasing chemical potential, these originally isolated orbitals will form local superfluid puddles that will eventually merge into a global superfluid phase.

Fig. 1: Lattice potential and sketch of possible phases.
figure 1

a, The 2D quasicrystalline optical lattice is generated by superimposing four independent 1D lattices in the x–y plane, marked by small arrows. A deep z lattice (large arrows) divides the system into a series of independent quasi-2D layers. b, Exemplary potential in a single layer. c, Repulsive interactions can delocalize an originally localized disordered system. From top to bottom, the sketches show the transition of the system’s ground state with increasing chemical potential μ, starting from the Anderson insulator (AI) in the non-interacting limit (μ = ϵ0 = 0), where the disorder strength Δ is above the critical disorder strength for localization Δc, over the localized but compressible Bose glass (BG) for weak repulsive interactions where bosons spill over into other low-lying minima and form local superfluid puddles, into the superfluid (SF) when the chemical potential is comparable to or larger than the disorder strength Δ.

Full size image

As the lowest-lying minima are typically located arbitrarily far away from each other, any changes or relaxation processes that require redistribution of particles between these distant minima may, in localized phases, require arbitrarily long times. In the non-interacting Anderson limit, orbitals localized at different local minima can indeed possess arbitrarily close energies while having only exponentially weak couplings15, thus resulting in many almost degenerate levels. This absence of level repulsion is a hallmark of non-ergodic phases16,17. As a consequence of these exponentially small gaps, even rather slow parameter changes within the Bose glass could be expected to take the system out of equilibrium.

In fact, although local changes within a localized system can induce changes over distances that are large compared with the localization length, the characteristic distance could be shown to increase only logarithmically with time and thus directly leads to exponentially large timescales in large systems18. Therefore, the thermodynamic notion of quasistatic or adiabatic changes, where the system remains in thermal equilibrium at all times and the process is isentropic, may not apply. This is reminiscent of many-body localization (MBL)2 and opens the question to which degree the Bose glass can be seen as the low-energy limit of a more general potential bosonic MBL phase.

Disordered interacting bosons have been studied for instance using helium-4 in porous media19, Cooper pairs in superconducting films20 and disordered quantum magnets21. In the context of ultracold atoms, the Bose glass has been extensively studied using various numerical methods22,23,24,25,26,27,28,29,30. Previous experiments in one dimension have shown the loss of coherence but were strongly affected by finite-temperature effects31,32,33,34,35 and experiments in three dimensions using speckle disorder studied momentum and quench responses36,37.

In this work, we investigate the low-energy states of a weakly interacting Bose gas in a two-dimensional (2D) eight-fold rotationally symmetric quasicrystalline optical lattice5. By analysing the momentum distribution of the system, we observe the Bose-glass-to-superfluid phase transition, and map out the phase diagram in the weakly interacting regime. Furthermore, our work experimentally establishes the non-adiabatic nature of the Bose glass, thereby highlighting its continuous connection to potential bosonic MBL phases at finite energy density7,8.

A 2D quasicrystalline optical lattice

Quasicrystals are long-range ordered yet not periodic38,39 and thereby represent a fascinating middle ground between order and disorder. In contrast to purely random potentials, where in one and two dimensions all single-particle eigenstates are localized for any non-vanishing disorder40, quasiperiodic potentials support a phase transition from extended to exponentially localized states at a finite potential depth41,42, thus providing an ideal platform for studying disorder-induced phenomena.

In our experiment, we load a degenerate Bose gas of about 1.2 × 105 potassium 39K atoms without discernible thermal fraction into a 2D quasicrystalline optical lattice using a 45-ms-long exponential ramp (Methods). The optical quasicrystal is formed by superimposing four independent blue-detuned one-dimensional (1D) lattices in the x–y plane at 45° angles, as depicted schematically in Fig. 1a. Each of these lattices is a 1D standing wave created by a retro-reflected laser beam at wavelength λlat = 725.4 nm. In addition, a deep lattice along the direction perpendicular to the plane (z axis) effectively slices the system into an array of 2D layers (see the grey disks in Fig. 1a). The resulting potential is given by

$$\begin{array}{ccc} & & V({\boldsymbol{\rho }}=\{x,y\},z)={V}_{0}\mathop{\sum }\limits_{i=1}^{4}{\sin }^{2}({{\bf{k}}}_{i}\cdot {\boldsymbol{\rho }}+{\phi }_{i})+{V}_{z}{\sin }^{2}({k}_{z}z),\\ & & {{\bf{k}}}_{i}\in \frac{2{\rm{\pi }}}{{\lambda }_{{\rm{l}}{\rm{a}}{\rm{t}}}}\left\{\left(\begin{array}{c}1\\ 0\end{array}\right),\frac{1}{\sqrt{2}}\left(\begin{array}{c}1\\ 1\end{array}\right),\frac{1}{\sqrt{2}}\left(\begin{array}{c}-1\\ 1\end{array}\right),\left(\begin{array}{c}0\\ 1\end{array}\right)\right\},\end{array}$$
(1)

where V0 and Vz denote the lattice depths, and ki and kz are the respective wavevectors (\(|{{\bf{k}}}_{i}|={k}_{z}={k}_{{\rm{l}}{\rm{a}}{\rm{t}}}=2{\rm{\pi }}/{\lambda }_{{\rm{l}}{\rm{a}}{\rm{t}}}\)) of the four 1D lattices in the x–y plane and the z lattice. The phase offsets ϕi are central to describe phasonic degrees of freedom and topological pumping in these potentials, but have no significant role in localization in large systems43.

Throughout this work, the depths of the horizontal lattices are varied in the range of V0 = 1–4 Erec and the z lattice is kept at Vz = 20 Erec, where \({E}_{{\rm{rec}}}={\hbar }^{2}{k}_{{\rm{lat}}}^{2}/(2m)\) is the recoil energy, ħ is the reduced Planck constant and m is the atomic mass. The deep z lattice provides a sufficiently strong vertical confinement so that interlayer tunnelling is negligible. As a consequence, atoms loaded into the lattice will be tightly confined to individual quasi-2D systems that show an eight-fold symmetric quasicrystalline structure, as depicted in Fig. 1b. A red-detuned dipole trap (Methods) provides an overall harmonic confinement and gives rise to an inhomogeneous density distribution (Fig. 2c).

Fig. 2: Bose-glass-to-superfluid transition.
figure 2

a, TOF images (9 ms TOF, 5 shots averaged) for different scattering lengths a at a fixed lattice depth of V0 = 2.8 Erec. Although the system is localized in the non-interacting and very weakly interacting cases, the appearance of sharp interference peaks for stronger interactions signals the emergence of long-range phase coherence, characteristic for the superfluid. b, Width of the central peak, distinguishing the coherent superfluid (light blue) from the incoherent Bose glass (dark blue). The dashed line is a guide to the eye indicating the detected phase boundary in the centre of the cloud \({V}_{{\rm{loc}}}^{(a)}\). It is identical to the line shown in the inset and in Fig. 3d. White points and error bars denote the QMC prediction from ref. 6 (Methods). Images in a correspond to the parameters marked by red diamonds. The inset shows the condensate fraction fc extracted from the same set of images, highlighting the coexistence of the two phases. c, Phase transition in an inhomogeneous system. The shaded Gaussian denotes the in-trap atomic density and the parabola represents the external trapping potential. For shallow lattices, the ground state is purely superfluid (left). At the non-interacting critical depth \({V}_{{\rm{loc}}}^{(0)}\), the Bose glass starts to appear at the low-density edge of the cloud where interaction effects are small (middle). With increasing lattice depth, the phase boundary gradually moves inwards until the entire cloud enters the Bose glass phase at \({V}_{{\rm{loc}}}^{(a)}\) (right).

Full size image

Even though the lattice depths used for the 2D quasicrystalline lattice are rather low, the physics of the system is nonetheless captured by a dedicated quasiperiodic Bose–Hubbard (QBH) model43, which in second quantization reads

$${\widehat{H}}_{{\rm{QBH}}}=\sum _{i}{{\epsilon }}_{i}{\widehat{a}}_{i}^{\dagger }{\widehat{a}}_{i}-\sum _{i\ne j}{J}_{ij}{\widehat{a}}_{i}^{\dagger }{\widehat{a}}_{j}+\sum _{i}\frac{{U}_{i}}{2}{\widehat{n}}_{i}({\widehat{n}}_{i}-1).$$
(2)

Here \({\widehat{a}}_{i}^{\dagger }\) (\({\widehat{a}}_{i}\)) is the bosonic creation (annihilation) operator on the ith lattice site, and \({\widehat{n}}_{i}={\widehat{a}}_{i}^{\dagger }{\widehat{a}}_{i}\) is the corresponding number operator. The Hamiltonian \({\widehat{H}}_{{\rm{QBH}}}\) is characterized by three site-dependent parameters, namely, on-site energies ϵi (neglecting the harmonic confinement), tunnelling energies Jij and on-site interactions Ui ∝ a, whose scale can be independently controlled by tuning the atomic s-wave scattering length a by means of a Feshbach resonance (Methods). We set \({{\epsilon }}_{0}:= \min \,{{\epsilon }}_{i}=0\) and use \(\varDelta := \max \,{{\epsilon }}_{i}\) as an intuitive measure of ‘disorder strength’, even though the modulation in Jij and Ui also influences the physics.

In the weakly interacting regime, systems described by the Hamiltonian \({\widehat{H}}_{{\rm{QBH}}}\) host a phase transition from Bose glass to superfluid, as illustrated in Fig. 1c. At strong interactions with U ≫ J, they furthermore host commensurate Mott insulators6,43; however, this regime is not probed in the current paper (Methods). In this strongly interacting regime, the term Bose glass was introduced to describe the phase emerging when the charge order of the Mott insulator vanishes for strong enough disorder Δ ≈ U (refs. 25,29). This regime shows the same phenomenology as the weakly interacting Bose glass, namely being a compressible, gapless, insulating phase without long-range coherence, and hence they both belong to the Bose glass phase.

Phase diagram

Our main observable to distinguish superfluid and localized states is the momentum distribution detected using time-of-flight (TOF) imaging, that is, by releasing the atomic cloud from all trapping potentials and imaging its density distribution after 9 ms of free expansion. This can be understood as a matter-wave diffraction experiment where waves originating on different lattice sites expand, overlap and then interfere. Analogous to diffraction experiments in optics and in periodic lattices44,45, the coherence length, which quantifies the range of spatial coherence between lattice sites, determines the width of the matter-wave interference peaks. A high-contrast interference pattern composed of sharp peaks indicates the presence of long-range phase coherence, the signature of the superfluid phase. Localized states with only short-range coherence, however, result in an incoherent broad momentum distribution.

Figure 2a shows a series of TOF images recorded for different scattering lengths at a fixed lattice depth of V0 = 2.8 Erec. At this lattice depth, the single-particle ground state is strongly localized46, and the measured momentum distribution at vanishing scattering length (top-left panel) correspondingly shows the broad momentum profile of a localized Anderson insulator. With increasing interactions, however, we observe the emergence of initially faint but sharp interference peaks, signalling the phase transition from the incoherent Bose glass to a superfluid in the high-density core of the cloud. The remaining broad background corresponds to the incoherent Bose glass at lower densities, where the critical lattice depth is lower and approaches the non-interacting limit.

To quantitatively study this transition at the high-density centre of the cloud, we choose an observable that can detect the presence of even a small superfluid component, namely, the full-width at half-maximum (FWHM) of the central peak. The FWHM, extracted from 2D Gaussian fits, provides an almost binary signature: if there exists a superfluid component, the FWHM corresponds to the width of the superfluid peak, which is dominated by the in situ cloud size47 (Methods). Only when the superfluid completely vanishes, the FWHM jumps to the width of the incoherent background (Methods and Extended Data Fig. 3).

The resulting phase diagram for the centre of the trap is shown in Fig. 2b and clearly reveals two distinct phases: the coherent superfluid at shallow lattices (light blue) turns relatively abruptly into the incoherent Bose glass (dark blue) at an interaction-dependent critical lattice depth \({V}_{{\rm{loc}}}^{(a)}\). At vanishing scattering length, the observed \({V}_{{\rm{loc}}}^{(0)}\) coincides with the known single-particle localization point at around \({V}_{{\rm{loc}}}^{(0)}=1.78(2)\,{E}_{{\rm{rec}}}\) (refs. 41,46) up to minor corrections (≲1a0, where a0 denotes Bohr's radius) stemming from the presence of weak residual interactions due to small dipole–dipole interactions48 and calibration uncertainties (Methods). With increasing scattering lengths, the critical lattice depth \({V}_{{\rm{loc}}}^{(a)}\) indicated by the dashed line shifts considerably towards deeper lattices, directly demonstrating that even weak repulsive interactions can significantly counteract localization. The observed transition agrees well with the recent quantum Monte Carlo (QMC) simulations for the ground state reported in ref. 6. These low localization thresholds also imply large tunnelling energies43 and hence a high resilience to temperature. Therefore, the expected effects11 of the finite experimental temperature (<20 nK; Methods) would be at most on the order of the QMC error bars.

As a complementary observable that highlights the inhomogeneous nature of the system, the inset of Fig. 2b shows the same phase diagram analysed in terms of the condensate fraction \({f}_{{\rm{c}}}:= {{\mathcal{N}}}_{{\rm{coh}}}\,/{\mathcal{N}}\), that is, the number of atoms in the sharp interference peaks \({{\mathcal{N}}}_{{\rm{coh}}}\) divided by the total atom number \({\mathcal{N}}={{\mathcal{N}}}_{{\rm{coh}}}+{{\mathcal{N}}}_{{\rm{incoh}}}\), where \({{\mathcal{N}}}_{{\rm{incoh}}}\) represents the population of the incoherent background (see Methods for details). The condensate fraction is high for shallow lattices and begins to slowly decrease after the lattice depth exceeds the non-interacting critical depth \({V}_{{\rm{loc}}}^{(0)}\) (see also Fig. 3e). This initially small downwards trend gradually becomes stronger, and the condensate fraction eventually reaches zero at the same critical depth \({V}_{{\rm{loc}}}^{(a)}\) extracted from the FWHM measurement (dashed line).

Fig. 3: Non-adiabaticity of the Bose glass.
figure 3

a, Condensate fraction in the 2D quasicrystal (normalized within each plot) as a function of lattice depth for different loading durations and scattering lengths. Although 15-ms ramps result in consistently lower condensate fractions, there is no consistent difference between 30 ms and longer ramps. b, FWHM of the central peak (wr) after a linear ramp of duration τ from the 2D quasicrystal into a regular 3D cubic lattice, where the ground state is a superfluid. The coloured circles correspond to different depths of the quasicrystalline potential V0 for a fixed scattering length of a = 10a0. For \({V}_{0} < {V}_{{\rm{loc}}}^{(10)}\) (blue circles), the initial state in the quasicrystal is superfluid and the final states show strong superfluid order for all explored ramp times. For a deep Bose glass at \({V}_{0} > {V}_{{\rm{loc}}}^{(10)}\) (red circles), in contrast, there is no initial coherence and only a very limited degree of phase coherence can be restored, demonstrating the absence of adiabatic evolution into and out of the Bose glass. c, An equivalent measurement for a Mott insulator in a regular 3D cubic lattice (Vx,y,z = 16 Erec, a = 150a0). Although the initial state also lacks coherence, it can be rapidly restored by ramping down the lattice depth in τ ≳ 2 ms. Insets in b and c show TOF images and OD denotes the optical density. d, Phase diagram showing wr for a slow ramp with τ = 15 ms highlighting three different regimes: a pure superfluid (light blue), an intermediate regime where superfluid and Bose glass coexist in the trap, and finally the pure Bose glass (darker blues). The transition into the pure Bose glass is consistent with the phase boundary extracted in Fig. 2b (dashed line). e, Comparing condensate fraction fc in the quasicrystal with wr for a = 23a0, demonstrating the consistency of all observations. The dashed line denotes the critical lattice depth \({V}_{{\rm{loc}}}^{(23)}\) extracted from the main diagram of Fig. 2b and the grey area indicates the intermediate regime where superfluid and Bose glass coexist. The solid lines in b, c and e are guides to the eye.

Full size image

The gradual decrease in the condensate fraction is consistent with the expected coexistence of superfluid and Bose glass in the system. This is the result of the inhomogeneous atomic density caused by the background harmonic dipole trap, as illustrated in Fig. 2c: when atoms are loaded into the lattice, the low-density edge of the cloud, where interaction effects vanish, will start to localize at the critical depth for non-interacting atoms \({V}_{{\rm{loc}}}^{(0)}\) (ref. 37). As we further increase the lattice depth, the phase boundary that separates the Bose glass from the superfluid core will slowly move towards higher densities until all atoms are ultimately in the Bose glass phase.

Absence of adiabaticity at the Bose glass transition

In typical quantum phase transitions between ergodic phases, for example, from superfluid to Mott insulator45, an important experimental check is whether the phase transition was crossed adiabatically, and thereby reversibly, or whether the observed loss of coherence results from irreversible heating, due to, for instance, rapid non-adiabatic changes that generate entropy. In the present case, however, the situation is potentially rather different, as the Bose glass is expected to be non-ergodic such that the thermodynamic notion of adiabatic changes may not apply.

To investigate this, we first study in Fig. 3a the effect of different lattice loading durations on the resulting condensate fraction. A too-rapid lattice ramp (15 ms) gives rise to considerable heating already in the superfluid regime, leading to lower condensate fractions compared with slower ramps. Once the loading duration exceeds 30 ms, it however becomes irrelevant and the condensate fraction is independent of the loading rate, demonstrating adiabaticity within the superfluid and consistent critical lattice depths \({V}_{{\rm{loc}}}^{(a)}\).

To study whether the phase transition was crossed adiabatically, we next try to restore superfluid coherence. Here we first load the atoms into the 2D quasicrystalline lattice in 45 ms, and then continuously transform the non-periodic lattice into a periodic simple-cubic three-dimensional (3D) lattice. This transformation is carried out by linearly ramping the depth of the x, y and z lattices to 8Erec over various durations τ while simultaneously reducing the depth of the remaining two diagonal lattices (Fig. 1a) to zero. The 3D cubic lattice was chosen as in this lattice the ground state is a superfluid with a finite critical temperature for condensation for all studied parameters49.

Figure 3b shows the FWHM of the central peak, wr, in the final periodic lattice for different ramp times τ at a fixed scattering length (a = 10a0), and the outcome highlights the fundamentally distinct behaviours of the superfluid and Bose glass phases. For \({V}_{0} < {V}_{{\rm{loc}}}^{(10)}\) (blue circles), the system remained superfluid during the entire sequence, and the ground state can adapt rapidly from a quasiperiodic extended wave to a periodic Bloch wave, as indicated by the sharp and narrow diffraction peaks for all ramp durations. For \({V}_{0} > {V}_{{\rm{loc}}}^{(10)}\) (red circles), however, where the system has entered the Bose glass regime, the initial state contains only very short-range coherence and hence results in a high wr. Furthermore, the measured wr remains significantly above that of the superfluid even for the slowest ramps explored in this measurement. This demonstrates that the system in this regime can restore only a very limited degree of phase coherence and thereby directly highlights the significant entropy production arising from traversing, that is, entering, ramping through and exiting the Bose glass. In combination, the above measurements demonstrate that despite the loading duration becoming irrelevant for sufficiently slow ramps, it remains impossible to traverse the Bose glass isentropically, that is, in a thermodynamically adiabatic fashion.

To show that the reduced coherence is not solely caused by dynamical effects such as Kibble–Zurek-type dynamics47 during too-fast final ramps, Figure 3c shows an equivalent measurement starting from a Mott insulator in a deep 3D simple-cubic lattice, where phase coherence is recovered by reducing the lattice potential to the same final depth as in the previous case. In this case, sharp interference patterns can be recovered already within 2 ms of ramp-down time, consistent with previous observations45,47. This contrast not only experimentally confirms that the incoherent localized phase we observe in the optical quasicrystal is distinct from a Mott insulator but also highlights that the inability to traverse the Bose glass adiabatically is rather distinct from the critical slowing down expected at conventional continuous phase transitions46,47. It is consistent with glassy dynamics in general and the expected non-ergodic nature of the Bose glass in particular.

Figure 3d shows the FWHM of the central peak (wr) after a slow final ramp of τ = 15 ms and demonstrates that the observed breakdown of adiabaticity indeed coincides with the transition into the Bose glass. This is further corroborated by the cuts shown in Fig. 3e: as more and more atoms localize and enter the Bose glass, not only does the condensate fraction decrease but also the coherence cannot be restored.

Conclusion

In this work, we experimentally study the 2D Bose glass in an optical quasicrystal with eight-fold rotational symmetry by probing the coherence properties of the system. We directly observe the phase transition between the Bose glass and the superfluid, in good agreement with QMC simulations6. In addition, we study the possibility to traverse the Bose glass adiabatically and always find significant entropy increases that are consistent with the expected non-ergodic character of the Bose glass. This paves the way for testing the connection between the Bose glass, MBL and glassy dynamics more generally. Quasicrystalline and quasiperiodic lattices offer a unique route to study MBL, as their long-range ordered nature can exclude conventional ergodic rare regions41,50 that are expected to destabilize MBL by seeding thermalization avalanches in real random systems51,52.

Methods

Experimental sequence

The experimental sequence begins with loading an almost pure Bose–Einstein condensate of about 1.2 × 105 39K atoms without discernible thermal fraction from a red-detuned crossed optical dipole trap (λdip = 1,064 nm, with trap frequencies (ωx, ωy, ωz) = 2π × (55, 43, 330) Hz) into a blue-detuned 2D quasiperiodic optical lattice (λlat = 725.4 nm). The initial temperature is bounded from above by Ti < 20 nK due to a conservative lower bound on the observed condensate fraction. Even neglecting that temperatures for weakly interacting bosons typically decrease when loading into a lattice, the resulting change in critical chemical potential Δμ due to the finite temperature is small according to ref. 11, that is, Δμ/μ < 2.5% or, equivalently, Δμ/(μ − ϵ0) < 20%, where \({{\epsilon }}_{0}:= \min \,{{\epsilon }}_{i}\). During the loading, the individual lattice depths are increased in 45 ms from 0 to their target values using exponential ramps with a time constant of 10 ms. The used target depths for the four horizontal lattices range within V0 = 1–4Erec while a fixed depth of Vz = 20Erec for the vertical z lattice ensures the formation of well-defined quasi-2D systems. After this ramp, the atoms are held in the quasicrystal for 10 ms. For imaging, we apply a short ‘booster stage’53 before we switch off all trapping potentials and record the matter-wave interference pattern by taking an absorption image after 9 ms TOF.

The booster stage consists of linearly increasing the potential depth of the horizontal lattices in 40 μs to a final depth of Vfinal = 6Erec. This stage is sufficiently short to not change the coherence properties of the system while providing a tighter on-site confinement and thereby not only enhancing the brightness of high-order diffraction peaks but also significantly reducing the heavy saturation on the central momentum peak (Extended Data Fig. 1a,b).

The interaction strengths Ui ∝ a are independently controlled by tuning the atomic s-wave scattering length (a) using the Feshbach resonance close to 403 G of the \(| F=1,{m}_{F}=1\rangle \) state in 39K (refs. 54,55). Here F denotes the total angular momentum and mF denotes the magnetic quantum number of the state. To ensure broadly comparable density distributions, the scattering length is initially prepared at a common finite value of a = 12a0 before the lattice loading starts and is then changed using a 20 ms linear current ramp to the desired value within a = 0–30a0 starting after the first 5 ms of the lattice ramp. Subsequently, the scattering length remains constant until being suddenly switched to a = 0a0 at the beginning of the TOF.

A periodic cubic 3D lattice can be produced by using only two orthogonal 1D lattices (x, y) out of the four in-plane 1D lattices indicated in Fig. 1 as well as the perpendicular z lattice. This was used as the final lattice in the attempt to restore superfluid coherence. For the final lattice depths in Fig. 3b–e, the ground state in the cubic lattice is a superfluid for all studied interactions. Although the cubic lattice is a priori only one of several possible choices, ramping into a periodic 3D lattice has the advantage that it results in an ergodic system where long-range coherence emerges below a finite critical temperature49.

Furthermore, the same cubic lattice geometry was also used for preparing the initial Mott insulating state in Fig. 3c, where the restoration of phase coherence is then carried out by employing a 16–8Erec linear ramp on all the three lattice axes simultaneously.

Coherence length and extraction of condensate fraction

In the TOF images, the width of the sharp diffraction peaks of the superfluid is dominated by the finite initial cloud size, which in combination with the finite TOF acts as an effective resolution limit for the measured momentum distribution47. Therefore, no significant broadening is expected as long as the coherence lengths in the superfluid part remains above 3– 5λlat (ref. 47). In the inhomogeneous system, the FWHM of the central peak (compare Fig. 2b) corresponds to this resolution-limited width as long as the k ≈ 0 peak of the superfluid remains visible atop the incoherent background of localized atoms. The FWHM jumps to the background width once the interference peaks have completely merged into the background, thereby giving rise to the sharp signature shown in Fig. 2b. This jump hence stems from the combination of the inhomogeneous system with the effective resolution limit imposed by the finite TOF and would not be present in a homogeneous system.

The condensate fraction fc is a complimentary observable that measures the fraction of coherent atoms and is evaluated for every shot according to \({f}_{{\rm{c}}}={{\mathcal{N}}}_{{\rm{coh}}}\,/{\mathcal{N}}\), where \({{\mathcal{N}}}_{{\rm{coh}}}\) is the population in the sharp interference peaks, and \({\mathcal{N}}={{\mathcal{N}}}_{{\rm{coh}}}+{{\mathcal{N}}}_{{\rm{incoh}}}\) is the total atom number with \({{\mathcal{N}}}_{{\rm{incoh}}}\) being the number of atoms in the incoherent background.

To extract \({{\mathcal{N}}}_{{\rm{coh}}}={\sum }_{k}{n}_{k}\) from the TOF images, we first identify the most pronounced 81 diffraction peaks within the first six diffraction orders5 and then extract their populations nk by fitting independent 2D Gaussian profiles to each peak. To prevent counting spurious populations from weakly populated peaks, we exclude fitted populations nk below 0.12% of the total atom number. Extended Data Fig. 1c illustrates the extracted populations.

The atom number in the incoherent background, \({{\mathcal{N}}}_{{\rm{incoh}}}\), is acquired by performing an additional 2D Gaussian fit to the whole cloud (region of interest \(3.3\times 3.3\,{(\hbar {k}_{{\rm{lat}}})}^{2}\)), where all detected diffraction peaks were masked during the fitting.

Parameter calibration

The two main experimental parameters are the lattice depth and the scattering length between atoms. The lattice depth is calibrated to within 0.1Erec by analysing the dynamics of Kapitza–Dirac diffraction for each 1D lattice individually; see the supplementary material of ref. 5 for details.

The scattering length is calibrated by observing the prominent atom-loss features corresponding to the zero-crossing of the scattering length, where the in situ density is highest, and the Feshbach resonance, where the loss coefficient is maximal. We then interpolate the scattering length between them using the common functional form55,56. As an independent cross-check, the magnetic field is calibrated using radio-frequency spectroscopy of the \(| F=1,{m}_{F}=-1\rangle \) to \(| F=1,{m}_{F}=0\rangle \) transition in 87Rb and converted to a scattering length using literature values for the parameters of the Feshbach resonance55,56. The two approaches agree to ≲1a0.

Comparing with QMC simulations

The QMC calculations reported in ref. 6 were performed as a function of the density n in a homogeneous system at fixed interaction strength g. As the main panel of Fig. 2b focuses on the phase transition in the centre of the trap, we extract the experimental central density n0 from in situ absorption images using the known aspect ratio of the trap. To minimize statistical noise, we measure n0 at different scattering lengths (a = 0–30a0) and constant lattice depth (V0 = 1Erec) and find a mild interaction dependence n0(30a0) ≈ 1/2 n0(0a0) for the used lattice ramp (Extended Data Fig. 2). In addition, we relate the 2D interaction coupling constant g used in ref. 6 back to the 3D scattering length a via

$$\begin{array}{c}g=\frac{{{\hslash }}^{2}}{m}\mathop{g}\limits^{ \sim }\,,\,\mathop{g}\limits^{ \sim }\approx {\mathop{g}\limits^{ \sim }}_{0}=\frac{2{\rm{\pi }}}{{\rm{l}}{\rm{n}}({a}_{{\rm{l}}{\rm{a}}{\rm{t}}}/{a}_{2{\rm{D}}})},\\ {a}_{2{\rm{D}}}=2.092\,{l}_{\perp }\exp \left(-\sqrt{\frac{{\rm{\pi }}}{2}}\frac{{l}_{\perp }}{a}\right).\end{array}$$

Here alat = λlat/2 and \({l}_{\perp }=\sqrt{\hbar /m{\omega }_{\perp }}\) is the characteristic confining length given by the strong z lattice with a trapping frequency of ω⊥ = 2π × 87 kHz.

Excluding Mott insulators

To investigate the possibility of Mott insulators in our experiment, we numerically compute the Bose–Hubbard parameters of the quasiperiodic potential using the results from ref. 43. We calculate the site-dependent ratio between on-site interactions and tunnelling energies \({U}_{i}/{\sum }_{j}| \,{J}_{ij}| \), where the sum runs over all significant tunnelling elements linking site i to other adjacent sites. Within the explored parameter regime, this ratio reaches a maximum of \(\max ({U}_{i}/{\sum }_{j}| \,{J}_{ij}| )\approx 1.4\) for the case of a = 30a0 and V0 = 4.0Erec. This is significantly below the critical interaction strength for forming a Mott insulator in a 2D square lattice (U/zJ)c ≈ 4.385 (ref. 57), where z = 4 represents the number of nearest neighbours. Furthermore, the studied parameter range lies within the weakly interacting regime of ref. 11, and Mott insulators can hence be excluded in this experiment.