Abstract
The disordered Bose–Hubbard model—a paradigm for strongly correlated and disordered bosonic systems^{1}—is central to our understanding of quantum phase transitions^{2}. Despite extensive theoretical work on the disordered Bose–Hubbard model, little is known about the impact of temperature, the dynamical behaviour of quantum phases, and how equilibrium is affected during quantum phase transitions. These issues are critically important to applications such as quantum annealing^{3,4,5,6,7} and electronics based on quantum phase transitions^{8}. Here, we use a quantum quench of disorder in an ultracold lattice gas to dynamically probe the superfluid–Bose glass quantum phase transition at nonzero temperature ( Fig. 1). By measuring excitations generated during the quench, we provide evidence for superfluid puddles in the Bose glass phase and produce a superfluid–Bose glass phase diagram consistent with completely constrained, finite temperature, and equilibrium quantum Monte Carlo simulations. The residual energy from the quench, which is an efficacy measure for optimization through quantum annealing, is unchanged for quench times spanning nearly a hundred tunnelling times.
Main
To make these measurements, we create an atomic realization of the threedimensional (3D) disordered Bose–Hubbard model (DBHM) using ultracold ^{87}Rb atoms trapped in a disordered optical lattice^{9}. The DBHM has been used to describe systems as diverse as ^{4}He in disordered media, Josephsonjunction arrays, and longwavelength properties of superconducting electron pairs. In the DBHM, a strongly interacting superfluid (SF) undergoes a quantum phase transition (QPT) into a Bose glass (BG) when subjected to disorder. The critical exponents of this transition fall in the percolation class^{10,11}. The BG phase exhibits the peculiar property of lacking longrange order while possessing infinite superfluid susceptibility^{1}, and it is therefore viewed as a gapless insulator with finite compressibility that arises from the presence of quasicondensates, or SF puddles, embedded in an insulating background. Disordered ultracold atom gases have been used to indirectly measure the SF–BG transition by means of transport and coherence measurements in 1D (ref. 12) and 3D (ref. 13) disordered lattices and in 1D quasiperiodic lattices^{14,15}.
In our experiment, we prepare a gas consisting of (27 ± 2) × 10^{3} atoms cooled below the critical temperature T_{c} for Bose–Einstein condensation in a parabolic trap such that the condensate fraction is greater than 90%. A disordered cubic optical lattice formed from pairs of counterpropagating λ = 812 nm laser beams and a 532 nm optical speckle field is superimposed on the gas (see Supplementary Methods). The atoms experience a potential energy shift proportional to the speckle intensity, which varies randomly in space, leading to disorder in the Hubbard parameters. The DBHM we realize is characterized by the Hamiltonian where i and j index the lattice sites, 〈〉 indicates that tunnelling occurs only between adjacent sites, and t_{ij}, U_{i} and ε_{i} are the Hubbard energies. In equation (1), is the number of particles on site i, () removes (adds) a particle from site i, m is the atomic mass, ω is the geometric mean of the trap frequencies, r_{i} is the distance to the centre of the trap, and μ is the chemical potential. We measure all energies in terms of the recoil energy E_{R} = h^{2}/2mλ^{2} ≈ k_{B} × 170 nK. The distribution of the Hubbard parameters, which are broadened around the values for the uniform system, are precisely known^{9,16}. The strength of the disorder is characterized by the average potential energy Δ associated with the speckle, which is approximately equal to the standard deviation of the ε_{i} distribution. The lattice potential depth s (which controls U/t) and Δ are independently adjusted by tuning the power of the lattice laser and 532 nm light. For the values of s sampled in this work, the gas is a strongly correlated, quantumdepleted SF when Δ = 0. We do not explore sufficiently high s to generate a Mott insulator phase in the gas.
We probe the BG–SF transition by measuring the amount of excitation produced by quenching Δ at fixed s. The disorder strength is linearly ramped from an initial value of Δ_{0} to zero in 30 ms (Fig. 1b), which is slow enough to avoid creating excitations solely through the time variation of the spatially inhomogeneous disorder potential (see Supplementary Methods). On the basis of general arguments regarding the phase diagram in untrapped systems, the BG phase will appear in the lowdensity edge of the gas for sufficiently high Δ_{0} (ref. 1). For stronger disorder, the BG–SF boundary moves inwards, encompassing more of the atoms. Excitations produced by the quench are measured using timeofflight (TOF) imaging. By imaging after a long (50 ms) period of free expansion, vortices and any other excitations involving velocity fields (including phonons) are transformed into modulations of the density profile and the measured optical depth (OD). These excitations are visible in the characteristic images shown in Figs 1c and 2b. For low Δ_{0}, the density profile after the quench and TOF is smooth, whereas for high Δ_{0}, features consistent with vortices are present.
To quantitatively characterize the amount of excitation present after the quench, we measurewhere is the measured OD at the pixel indexed by i and j, and f is a smooth fitting function (see Supplementary Methods). This method was previously used to observe the quantum Kibble–Zurek (KZ) effect by measuring excitations generated through a quench between Mott insulator (MI) and SF states in a ‘clean’ lattice^{17}. Data for s = 11E_{R} and Δ_{0} ≈ 0–1E_{R} are shown in Fig. 2a. It is apparent that excitations are not generated by the quench until a threshold disorder strength is crossed, above which increases approximately linearly with Δ_{0}. Similar threshold behaviour is observed for all s we sample in this work.
One way to account for such behaviour is by excitations created through SF puddles merging as the BG–SF boundary is crossed^{18}. As spatially distinct SF puddles combine, the random phases associated with each island naturally lead to vortices and other excitations. This process and the threshold behaviour evident in Fig. 2a is generally described by the quantum KZ effect in the context of QPTs (ref. 19). In the KZ scenario, an adiabatic transition from a phasedisordered (for example, BG) to an ordered (for example, SF) state is impossible because of diverging characteristic scales of length and time in the vicinity of the transition. Dynamically traversing a continuous QPT by tuning (or quenching) a Hamiltonian parameter leads to the formation of excitations that persist even after the transition is crossed. The quantum KZ effect has been used to describe this phenomenon, and it has been observed for the MI–SF transition in a ‘clean’ optical lattice, where sensitivity to the equilibrium phase boundary and powerlaw scaling were observed^{17}, and the scaling of the coherence length after the quench with the quench rate was measured^{20}. Our knowledge of how KZ physics is modified by disorder and glassy phases is limited to simulations and theory of certain onedimensional spin models^{3,21,22}. The prime example of a quantum quench in a disordered system before this work was in an Ising magnet^{23}, where a smaller residual energy for quantum annealing compared with thermal annealing was observed by means of magnetic susceptibility measurements.
To connect the observed threshold disorder with the SF–BG transition, we carry out exact quantum Monte Carlo (QMC) simulations of the equilibrium system using the same trap and lattice parameters, atom number, and speckle disorder as in the experiment (see Supplementary Methods). For trapfree geometries in the thermodynamic limit, the BG is characterized by a vanishing superfluid order parameter and nonzero compressibility^{24}. In contrast, the trapped system we consider exhibits domains corresponding to SF and BG phases that we distinguish using the spatial extent of the condensate. The condensate is identified as the macroscopic occupation of a singleparticle eigenstate that we can obtain from the singleparticle density matrix (refs 25,26).
For clean systems (that is, Δ = 0) with U/t < 29.34 ± 0.02 that are below T_{c}, a single condensate extends throughout the system that coincides with the local superfluid density order parameter (see Supplementary Methods). As Δ is increased, this behaviour changes, and the extent of the macroscopic condensate shrinks, leaving behind regions devoid of coherence. Because the SF–BG transition is of the continuous type, phase coexistence is forbidden, and we identify these regions as BG phase. To illustrate how the BG phase emerges in the gas, we show the two highest occupation eigenfunctions of ρ_{1} for s = 11E_{R} and Δ = 0.05E_{R} and 1E_{R} in Fig. 2c. At low Δ, all singleparticle states are spatially overlapped with the SF domain, and the second highest occupied state results from interactioninduced quantum depletion. For sufficiently high Δ, however, this extended state is replaced by a spatially localized mode that corresponds to a nonmacroscopic and locally coherent superfluid puddle, characteristic of the BG phase.
To compare with the measurements, we compute the BG fraction N_{BG}/N as the fraction of atoms in regions without a macroscopic condensate present. This estimate is an upper bound at nonzero temperature because of thermal excitation, which we find is small in the regime we study (see Supplementary Methods). As shown in Fig. 2d, the BG as defined by this criterion emerges at the edge of the gas and grows in extent and number as Δ is increased. The typical behaviour for N_{BG}/N at s = 11E_{R} as Δ is varied (shown in Fig. 2c) mirrors the amount of excitation created by the quench in the experiment: N_{BG}/N is only nonzero above a threshold disorder, above which it increases approximately linearly with Δ.
We construct a SF–BG phase diagram (Fig. 3) by estimating the threshold disorder Δ_{th} for excitations to appear in the experiment and for BG to appear in QMC simulations using a piecewiselinear fit to data such as those shown in Fig. 2. The fitting function assumes constant behaviour for disorder strengths less than Δ_{th}, and linearly increasing behaviour characterized by the free parameters Δ_{th} and a slope for disorder strengths greater than Δ_{th}. Several important features of the phase diagram are evident. The threshold disorder Δ_{th} is weakly dependent on U/t, and the QMC and experimental results agree within the 25% systematic uncertainty in Δ_{0}; there are additional systematic and statistical uncertainties arising from finite temperature and disorder averaging (see Supplementary Methods). This agreement demonstrates that the quench dynamics and production of excitations in this strongly disordered system are sensitive to the groundstate, equilibrium phases (which are not significantly distorted by nonzero temperature), and that the QMC simulations accurately describe the physical experiment. Moreover, it implies that the SF puddles present in the QMC simulations (Fig. 2c) exist in the experiment. The observed threshold behaviour cannot be explained by mean field theory, which predicts that a BG appears for infinitesimal disorder^{27}. The decrease in Δ_{th} at higher s (that is, larger U/t), which cannot be accounted for by general classical percolation mechanisms, indicates that stronger interactions weaken the SF against localization in this regime. Finally, the observed Δ_{th} is lower than that required to convert the entire gas to a localized BG phase, as determined through transport measurements^{13}. This behaviour is expected, because the entire gas must be converted to a BG phase in order to halt transport.
We explore the dynamical timescale of the SF–BG transition by varying the quench time τ_{Q} at fixed s and Δ_{0}. In the KZ scenario for clean systems, the amount of excitation and heat produced during a quantum quench typically exhibit powerlaw dependences on the quench time^{19}. The knowledge of how this changes in disordered systems is restricted to theory and simulations of onedimensional spin chains, which possess an inverse logarithmic dependence of the residual energy (that is, the energy generated by the quench) on the quench time^{21,22}. Data for nearly two orders of magnitude in τ_{Q}/(ℏ/t), where ℏ/t is the tunnelling time, are shown in Fig. 4 at s = 10E_{R} for the BG regime at Δ_{0} = 1E_{R}. To avoid complications from decay of excitations during the quench, we measure the temperature of the gas after allowing rethermalization in the trap for 150 ms (see Supplementary Methods). We show the residual energy as the change in the thermal energy δε = ε − ε_{0}, where ε and ε_{0} are the thermal energy per particle inferred from the measured temperature with and without the quench.
The extremely weak dependence of δε on τ_{Q} is characteristic of the BG regime for all s sampled in this work. The residual energy is nearly fixed for quenches spanning approximately six to sixty times the tunnelling time. We do not observe significant changes in this behaviour with different s. The range of Δ_{0} we can sample is limited to approximately 1E_{R} at s = 10E_{R}—and less at higher s because this measurement requires the heat generated by the quench to be greater than that from the lattice light, and for the gas after the quench to remain condensed. Constraining the scaling with τ_{Q} is challenging given the observed weak dependence and our inability to explore longer quench times because scattering of the lattice light heats the gas above T_{c}. The fits to an inverse logarithm proportional to 1/log(τ_{Q}t/ℏ)^{1.0±0.4} and a power law (τ_{Q}t/ℏ)^{0.4±0.1} shown in Fig. 4 are equally consistent with the data.
Theoretical predictions for how the residual energy changes with quench time in this system are unavailable, as exact numerical simulation of a quench is intractable for experimentally relevant numbers of particles in two and three dimensions. Approximate simulations of other dynamics have been carried out, including exploration of how modulating the t leads to excitations^{28}. A clearer understanding of precisely how the residual energy in disordered quantum systems scales with quench time is critical to applications such as quantum annealing, which can be used for many NPhard optimization problems that map onto disordered quantum systems^{5,7,29}.
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Acknowledgements
The authors acknowledge funding from the National Science Foundation (grants PHY 1205548 and PHY 1505468) and the Army Research Office (grant W911NF1210462). Computation time was provided by XSEDE resources at TACC (Texas) and INCITE resources at Oak Ridge National Laboratory.
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 Carolyn Meldgin
 & Ushnish Ray
These authors contributed equally to this work.
 Ushnish Ray
Present address: Department of Chemistry, Princeton University, Frick Chemistry Laboratory, Princeton, New Jersey 08544, USA.
Affiliations
Department of Physics, University of Illinois at UrbanaChampaign, Urbana, Illinois 61801, USA
 Carolyn Meldgin
 , Ushnish Ray
 , Philip Russ
 , David Chen
 , David M. Ceperley
 & Brian DeMarco
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Contributions
B.D., C.M., U.R. and D.M.C. conceived the research. C.M. and U.R. contributed equally to this work: C.M. conducted and analysed the measurements, and U.R. performed and analysed the numerical simulations. P.R. and D.C. contributed to the measurements and data analysis. B.D. and D.M.C. supervised the experimental and theoretical work, respectively. B.D., U.R. and C.M. wrote the manuscript, which was discussed by and commented on by all authors.
Competing interests
The authors declare no competing financial interests.
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Correspondence to Brian DeMarco.
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