Abstract
Ultracold atomic gases have revolutionized the study of nonequilibrium dynamics in quantum manybody systems. Many counterintuitive nonequilibrium effects have been observed, such as suppressed thermalization in a onedimensional (1D) gas^{1}, the formation of repulsive selfbound dimers^{2}, and identical behaviours for attractive and repulsive interactions^{3}. Here, we observe the expansion of a bundle of ultracold 1D Bose gases in a flatbottomed optical lattice potential. By combining in situ measurements with photoassociation^{4,5}, we follow the spatial dynamics of singly, doubly and triply occupied lattice sites. The system sheds interaction energy by dissolving some doublons and triplons. Some singlons quantum distil out of the doublon centre^{6,7}, whereas others remain confined^{7}. Our Gutzwiller meanfield model captures these experimental features in a physically clear way. These experiments might be used to study thermalization in systems with particle losses^{8}, the evolution of quantum entanglement^{9,10} or, if applied to fermions, to prepare very low entropy states^{6}.
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Quantum distillation is a previously unobserved phenomenon in which atoms at singly occupied lattice sites (singlons) escape the central region of an untrapped lattice gas, leaving doubly occupied lattice sites (doublons) behind (Fig. 1a)^{6}. It depends on one readily achievable condition, that there be an energy mismatch that prevents isolated doublons from disintegrating into two singlons. Singlons in a sea of doublons can be understood as vacancies. The tunnelling rate of a singlon in an empty lattice is J and the bosonic vacancy tunnelling rate is 2J, because the vacancy moves when either of the adjacent doublon’s atoms tunnels. When bosonic vacancies reach a doublon sea edge, only those with intermediate energies can transmit into the empty lattice while conserving energy^{7}, as illustrated in Fig. 1b. Vacancies with quasimomenta outside that limited range reflect from the edge, confining them in the doublon sea. When a singlon does exit, it purifies and shrinks the doublon sea by one lattice site. It has been suggested that collisions of vacancies with triplons can thermalize the vacancies, possibly transferring them into transmissible quasimomentum states^{7}. In contrast, fermionic vacancies have the same tunnelling energy as singlons in the empty lattice, so they always pass through the edge of the doublon sea.
A previous nonequilibrium experiment with bosons in flat 1D lattices mostly focused on initial singleatom number states in a deep lattice, and studied the expansion dynamics after a quench of the onsite interaction energy^{3}. Such a quench leaves the manybody wavefunction out of local equilibrium. In contrast, we start our experiments with trapped, superfluid, 1D Bose gases in lattices with an average of between one and two atoms per site, and we quench by suddenly removing the trap. This is a fundamentally different quench, a geometric quench, after which the manybody wavefunction is still locally in equilibrium. Geometric quenches have been theoretically shown to lead to remarkable universal phenomena, such as quasicondensation at finite momenta^{11,12}, dynamical fermionization of the Tonks–Girardeau gas^{13,14}, and identical expansions of bosons and fermions^{15}. Our experimentally observed expansion dynamics are qualitatively reproduced by a Gutzwiller meanfield calculation (see Methods). By theoretically and experimentally studying the spatial evolution of site occupancy, we obtain a straightforward physical interpretation of the dynamics. It exhibits clear signatures of quantum distillation^{6} and confinement^{7,16} of vacancies in the doublon sea.
In our experiment, Bosecondensed ^{87}Rb atoms in a crossed dipole trap are slowly loaded into an array of 1D tubes formed by a bluedetuned 2D optical lattice (wavevector k = 2π/773 nm), with a superposed axial optical lattice of variable depth V_{0} and a reddetuned crossed dipole trap for overall confinement (see Methods). For the density and V_{0} used in this work, the initial ground states are predominantly superfluid^{17}. As each lattice site starts with a superposition of number states, pictures such as those in Fig. 1a represent one of many distributions whose coherent sum is the state of the system. As we will see, most of the qualitative behaviour of what are thus delocalized singlons and doublons can be understood using localized pictures, leavened by the understanding that each picture represents only a small piece of the overall wavefunction.
At t_{ev} = 0, we suddenly lower the depth of the crossed dipole trap, leaving enough power to cancel the residual antitrap due to the 2D lattice beams over a range of ∼160 μm (see Methods). We observe the subsequent spatial evolution in three ways (referred to as M1, M2 and M3), which together allow us to separately determine the spatial evolution of the probability distributions of singlons, doublons and atoms at more highly occupied sites. In M1, we measure all the atoms after a given time t_{ev} by switching to a 1D axial lattice of depth 27E_{rec} (where the recoil energy E_{rec} = ℏ^{2}k^{2}/2M and M is the Rb mass) and allowing the atoms to expand radially so that the density is low enough for absorption imaging. Information about the transverse distribution among tubes is lost, but the axial distribution is preserved, with a resolution of ∼3 μm. In M2, at t_{ev} we suddenly switch to a 27E_{rec} lattice in each of three directions and turn on a photoassociation pulse for 1.5 ms (refs 4, 5), which is long enough to eliminate all doublons, 2/3 of the triplon atoms, and most of the atoms from sites with higher occupancies. The axial distribution measurement is then made as in M1. In M3, at t_{ev} we suddenly switch to the 27E_{rec} 3D lattice and wait for 50 ms, which is long enough for threebody inelastic collisions^{18,19} to empty the triplon sites and most atoms from more highly occupied sites. The axial distribution is then measured as in M1.
Figure 2a–c shows the evolution of the total atom distribution (M1) for V_{0} = 3E_{rec}, 4E_{rec} and 5E_{rec}. These depths correspond to U/J of 4.7, 6.8 and 9.6 (ref. 20), respectively, in the oneband Hubbard model^{21,22}, where U is the onsite repulsion energy. All are characterized by a central core of atoms that steadily releases atoms that tunnel away from the centre. Figure 2d–f shows the distributions of single atoms, which are derived from M1, M2 and M3 (see Supplementary Fig. 1a–f). These curves show two dominant features. First, the cores contain many single atoms. Second, the broader pedestals of the M1 distributions are composed nearly exclusively of singleatom sites. The velocities of the leading edges of the pedestals equal, to within ∼10% systematic uncertainties, the calculated maximum possible velocity (v_{max} = 2Ja/ℏ, where a = λ/2 is the lattice spacing) for single atoms tunnelling in the lowest band (see insets in Fig. 2a–c). These velocities start to decrease at the end of the compensation range of the crossed dipole trap, ultimately Bragg scattering backwards; we do not show data after atoms return to the core. Figure 2g–i shows the distribution of doublons, derived from all three measurement types. The number of doublons decreases steadily after the quench, but the widths of the doublon distributions barely change. The triplon distributions (see Supplementary Fig. 1g–i) have the same width as the doublon distributions to within a 10% uncertainty.
Figure 2j–l (and Supplementary Fig. 1j–l) shows the results of a Gutzwiller meanfield calculation^{16,23} (see Methods), which simulates an array of identical tubes with different atom numbers, as in the experiment, and discretizes the direction along the tubes^{24} so that the standard singleband approximation is not used. The theory assumes an initial zero temperature Bose–Einstein condensate. This means that finite temperature and quantum fluctuations due to the 1D character of the system are not taken into account. The initial distribution is thus an imperfect match to the experiment (see Methods). The doublons initially expand farther in the theory than in the experiment, presumably because of the longrange initial phase coherence in the theory. The early decrease in the theory’s doublon density no doubt affects the details of the ensuing dynamics; however, qualitatively, the theory behaves like the experiment in all respects other than the shape and size of the doublon distributions. For further comparison to theory, see Supplementary Information.
Figure 3a–c shows the number of singlons, doublons and triplons as a function of time, derived from the appropriate combinations of M1, M2 and M3. The numbers of doublons and triplons drop steadily, with corresponding increases in the numbers of singlons. The theory shows similar behaviour (see Fig. 3d). Although conservation of energy dictates that isolated doublons cannot dissociate for U/J ≳ 4 (ref. 2), in a predominantly doublon sea the aforementioned tunnelling enhancement doubles this limit. Similarly, one can show that a sea of singlons increases the limit by 50%. That our doublons live in a bath intermediate to these two seas explains why they dissociate, at least for U/J ≲ 8, and why the dissociation rate decreases at long times (see especially Fig. 3a after 50 ms) when the number of empty sites in the centre increases. Dissociation for U/J = 9.6 (V_{0} = 5E_{rec}) naively requires that energy be shared among more singlons^{25,26}, but it might be that the oneband Hubbard model calculation^{20} overestimates the effective value of U/J. The latter interpretation is supported by the fact that, at short times, the evolutions of all the curves in Figs 2 and 3 are approximately selfsimilar when the time axes are multiplied by J (see Supplementary Fig. 2), to within the small differences in the doublon distribution widths discussed below. Although the physics is dominated by interacting particle effects, marginal changes in site occupancy scale with J. Our meanfield calculation allows us to explicitly track the conversion of potential energy (interaction + lattice potential) into kinetic energy (see inset) that results primarily from doublon dissolution.
Quantum distillation is difficult to isolate at early times, because it occurs while initially unconfined singlons are also leaving the central region and singlons are being created by dissolution. But quantum distillation dominates at 5E_{rec} (Fig. 2f) after 20 ms, by which time the doublon number is stable (see also Figs 2i and 3c) and the unconfined singlons present a locally flat background. The number of singlons confined in the doublon sea as a function of time is plotted in Fig. 4a (see Methods). Its steady decrease is a clear signature of quantum distillation, further supported by the fact that the rate scales with J (see also Supplementary Information). At late times, the fraction of confined singlons levels off at ∼5%, showing longterm vacancy confinement in the doublon sea. The meanfield calculations also show some singlons initially leaving and others remaining indefinitely, but fewer singlons distil out in the calculations. This is expected because the real 1D gas has a broader initial quasimomentum distribution. Thus, in the calculation there are more singlons with the lower energies that do not transmit out of the doublon sea.
In Jrescaled time, after doublons stop dissolving at 5E_{rec} they are still dissolving at lower lattice depths. That the three sets of data points in Fig. 4a overlap means that extra doublon dissolution does not affect the central singlon number. This could be because the vast majority of singlons created when doublons dissolve have the right quasimomentum for immediate quantum distillation, and leave the centre rapidly.
Further evidence of quantum distillation is given in Fig. 4b, which shows the evolution of the full width at half maximum (FWHM) of the doublon distribution. There are three sizechanging processes, each dominating for a time. The FWHMs increase during the first 10–20 ms because doublons can initially expand into singly occupied sites and perhaps there is more doublon dissolution in the middle (see Fig. 2g–i). The FWHMs then decrease owing to the mechanics of distillation, where escaping singlons move the last doublon one site inward. When the rate of quantum distillation decreases (near 4 ms ⋅ E_{rec} in Fig. 4a) then, as long as the lattice depth is small enough, shrinking is overtaken by expansion. We suspect this to be the result of higherorder processes involving confined singlons that compromise the stability of the edges of the doublon sea^{27}. This unanticipated higherorder effect, undoubtedly absent in fermions and not present in the meanfield theory results, further limits the effectiveness of bosonic quantum distillation in producing lowentropy blocks of doublons. The approximate stability or slight increase in the width of the doublon distribution implies that, as time evolves, the number of empty sites among the doublons increases.
Our work has concentrated on spatial distributions, which are local properties, but it should also be possible to use related techniques such as timeofflight measurements to study nonlocal properties, such as quasimomentum distributions and correlations. This simple lattice system, in which the doublon sea is open but nonetheless settles to a stable steady state, can help address major open questions in quantum dynamics, such as how systems thermalize in the presence of particle losses^{8}. Studying how quantum correlations grow after the quench should give a more general insight into how entanglement spreads in quantum systems^{9}. The fact that our meanfield treatment (exact only in infinite dimensions) qualitatively captures the 1D dynamics, suggests that similar dynamics occur in higher dimensions. Finally, an experimental implementation with fermions holds the promise of producing doublon cores with superlatively low entropy^{6}, which might allow the study of hitherto inaccessible models of quantum magnetism^{28} and hightemperature superconductivity^{29}.
Methods
Experiment.
Trapping: We start with 2 × 10^{5} Bosecondensed ^{87}Rb atoms in the F = 1, m_{F} = 1 state in a crossed dipole trap^{30} made from 1.064 μm laser beams, with 1.8 W per beam and 160 μm beam waists. Gravity is cancelled by a magnetic field gradient. A bluedetuned 773.5 nm wavelength 3D optical lattice, made from two retroreflected horizontal 450 μm waist beams and a retroreflected vertical 700 μm waist beam, is turned on in 14 ms to a depth of V_{0}, after which the two horizontal lattice beam pairs are increased to their full depth of 40E_{rec} in 44 ms. We find that the results of the experiment (including spatial distributions and occupancy fractions) do not change significantly as long as the lattice turnon times are 35 ms or longer. That remains true if we wait for tens of ms in all the traps before starting the evolution.
Flat lattice: As for technical reasons the lattice waists are much larger than the dipole trap waists, we can create a flat lattice only near the centre of the trap. To finetune the cancellation of the two potentials, we start with trapped 1D gases with no axial lattice, and choose the highest crossed dipole beam intensity at which no atoms remain trapped in the central region. This gives a central potential that is flat to within 0.08E_{rec} over a length of 160 μm.
Small bump: The small bump on the left of the initial atom distributions in the experimental results of Fig. 2 is due to spatial imperfections on the beams that make up the confining crossed dipole trap, where the atoms are trapped 3.4 Rayleigh lengths from the beam focus. After t_{ev} = 0 the atoms evolve in a much smoother potential, so only the initial distribution is affected by this issue, not the evolution. The bump can serve as a feature, because it provides confirmatory evidence of the fraction of singlons that remain confined (see, for example, the longtime curves in Fig. 2f).
Determination of the trapped singlon fraction: We analyse the singlon distributions (see Fig. 2d–f), by first determining the FWHM of the doublons (see Figs 2g–i and 4b) at each time. We then measure the difference between the peak of the singlon distribution and its value at the doublon FWHM positions. We assume that the confined peak height is twice that difference, and has the doublon width. We do not use this procedure at very early times, before there is a discernible shoulder in the singlon distribution, although the curves in Fig. 4a change little if we add a few earlier points. The assumption that the FWHMs of doublons and trapped singlons are the same is not exactly true for a bundle of tubes. This procedure also assumes that the unconfined singlon distribution is approximately flat in the central region, which is also not exactly true. Our confidence in the reliability of this procedure is buttressed by the universality of the curves in Fig. 4a, which holds despite the systematic differences in the doublon FWHMs at different V_{0} (Fig. 4b).
Theoretical calculations.
For the theoretical analysis of the expansion, we model a collection of independent 1D tubes. To account for possible higherband effects during the dynamics, which may be important for the lowest lattice depths, we do not use the oneband approximation that results in the standard Bose–Hubbard model. Instead, we discretize the space along the tubes by introducing an artificial grid of spacing ℓ ≪ λ (we take ℓ = 0.05λ, where λ is the wavelength of the optical lattice) to obtain a representation of the continuum in terms of an artificial 1D Bose–Hubbard model^{24}. The particles are then subject to the periodic potential generated by the optical lattice and to the confining potentials generated by the crossed optical dipole trap. We simulate arrays of up to 110 × 110 tubes, with each tube’s length being 500λ. We choose the theoretical initial conditions so that the fraction of singlons in the simulation matches the experiment. This is done by arbitrarily tuning the strength of the t_{ev} < 0 axial and transverse trapping potentials, keeping their ratio fixed to the experimental value. The parameters for t_{ev} ≥ 0 are those in the experimental setup, except that the overall confining potential is set exactly to zero. The calculations are carried out within the Gutzwiller meanfield approximation^{16,23}, where the initial state is selected to be the ground state in the absence of tunnelling between the 1D tubes. Our results are robust to further reduction of the value of ℓ. For further details, see Supplementary Information.
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Acknowledgements
We are indebted to A. Daley, M. Fleischauer and F. Essler for illuminating conversations. This work was supported by NSF Grant No. PHYS167830, the Army Research Office, and the Office of Naval Research (J.C. and M.R.). J.C. acknowledges support from the John Templeton Foundation. Research at Perimeter Institute is supported through Industry Canada and by the Province of Ontario through the Ministry of Research and Innovation.
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Xia, L., Zundel, L., Carrasquilla, J. et al. Quantum distillation and confinement of vacancies in a doublon sea. Nature Phys 11, 316–320 (2015). https://doi.org/10.1038/nphys3244
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DOI: https://doi.org/10.1038/nphys3244
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