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Quantum distillation and confinement of vacancies in a doublon sea

Ultracold atomic gases have revolutionized the study of non-equilibrium dynamics in quantum many-body systems. Many counterintuitive non-equilibrium effects have been observed, such as suppressed thermalization in a one-dimensional (1D) gas, the formation of repulsive self-bound dimers, and identical behaviors for attractive and repulsive interactions. Here, we observe the expansion of a bundle of ultracold 1D Bose gases in a flat-bottomed optical lattice potential. By combining in situ measurements with photoassociation, 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 distill out of the doublon center, while others remain confined. Our Gutzwiller mean-field model captures these experimental features in a physically clear way. These experiments might be used to study thermalization in systems with particle losses or the evolution of quantum entanglement, or if applied to fermions, to prepare very low entropy states.

be used to study thermalization in systems with particle losses, 8 to study the evolution of quantum entanglement, 9,10 or if applied to fermions, to prepare very low entropy states. 6 Quantum distillation is a previously unobserved phenomenon in which the entropy in an untrapped lattice gas migrates to its edges and escapes with single atoms, leaving behind a colder metastable insulating core. 6 It results from the dynamics of atoms at singly occupied lattice sites (singlons), which act as vacancies in an array of doubly occupied lattice sites (doublons). It depends on one readily achievable condition, that there be an energy barrier that prevents isolated doublons from disintegrating into two singlons. Figure 1a shows a diagram of quantum distillation. Vacancies tunnel in the doublon sea, and can tunnel away as singlons when they reach the edge, thus purifying and shrinking the doublon sea. The tunneling rate of a singlon in an empty lattice is J and the bosonic vacancy tunneling rate is 2J, since either of the adjacent doublon's atoms can tunnel. When bosonic vacancies reach the 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 and get confined in the doublon sea. It has been hypothesized 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 tunneling energy as singlons in the empty lattice, so they always pass through the edge of the doublon sea. Relevant ground energy bands. Only atoms with quasimomentum between 1/3 and 2/3 of the band edge can tunnel from the doublon sea into the empty lattice. Outside that range transmission into the empty lattice does not conserve energy. This limitation on quantum distillation is absent for fermions.
A previous non-equilibrium experiment with bosons in flat 1D lattices mostly focused on initial single-atom number states in a deep lattice, and studied the expansion dynamics after a quench of the on-site interaction energy. 3 Such a quench leaves the many-body wavefunction out of local equilibrium. In contrast, we start our experiments with trapped, superfluid, 1D Bose gases in lat-tices 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 many-body wavefunction is still locally in equilibrium. Geometric quenches have been theoretically shown to lead to remarkable universal phenomena, such as quasi-condensation 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 mean-field 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, Bose condensed 87 Rb atoms in a crossed dipole trap are slowly loaded into an array of 1D tubes formed by a blue-detuned 2D optical lattice (wavevector k=2π/773 nm), with a superposed axial optical lattice of variable depth V 0 and a red-detuned crossed dipole trap for overall confinement (see Methods). For the density and V 0 used in this work, the initial ground states are expected to be predominantly superfluid. 17 At t ev = 0, we suddenly lower the depth of the crossed-dipole trap, leaving enough power to cancel the residual anti-trap 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 t ev by switching to a 27E rec deep 1D axial lattice (where the recoil energy E rec = 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, 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 three-body 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.  which simulates an array of identical tubes with different numbers, and discretizes the direction along the tubes 24 so that the standard single-band approximation is not used. The theory assumes an initial zero temperature BEC. This means that finite temperature and quantum fluctuations due to the one-dimensional character of the system are not taken into account. The initial distribution is thus an imperfect match to the experiment (see Methods). The initial expansion of the doublon distribution in the theory greatly exceeds that in the experiment, presumably because of the long range initial phase coherence in the theory. The early decrease in the theory's doublon density no doubt affects the details of the ensuing dynamics, but qualitatively, the theory behaves like the experiment in all respects other than the shape and size of the doublon distributions.   In conclusion, we have studied the dynamics of low temperature 1D Bose gases in flat bottom lattice potentials, keeping track of the spatial distribution of singly, doubly, and triply occupied lattice sites. Our work has concentrated on spatial distributions, which are local properties, but it should also be possible to use related techniques to study non-local properties, like 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 more general insight into how entanglement spreads in quantum systems. 9 The fact that our mean-field treatment (only exact in infinite dimensions) qualitatively captures the 1D dynamics, suggest that similar dynamics occur in higher dimensions. Finally, an experimental implementation with fermions holds the promise of producing superlatively low entropy doublon cores, 6 which might allow the study of hitherto inaccessible models of quantum magnetism 28 and high temperature superconductivity. 29,30 Methods Experiment Trapping: We start with 2×10 5 Bose condensed 87 Rb atoms in the F = 1, m F = 1 state in a crossed dipole trap 31 with 1.8 W per beam and 160 µm beam waists. Gravity is canceled by a magnetic field gradient. A blue-detuned 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  Figure 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 Figure 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 quite true. Our confidence in the reliability of this procedure is buttressed by the universality of the curves in Figure 4a, which holds despite the systematic differences in the doublon FWHMs at different V 0 (see Figure 4b).
Extended Data Figure 1 Extended Data Figure 1 V 0 =3E rec , 4E rec , and 5E rec respectively. The curves are approximately self-similar up until ~3ms-E rec , which suggests that doublon decay is predominantly a first-order process. The largest difference in the curves is that when the lattice is deeper, the doublons stop decaying at an earlier time, leaving a larger asymptotic doublon population.

Theory
We assume that the system under consideration can be described in terms of particles moving in an array of independent 1D tubes. This is justified because the tunneling amplitude between adjacent tubes is exponentially small in the strength of the optical lattice defining the array, which in the experiment is V transverse /E rec = 40. The Hamiltonian in each 1D tube is where the field operators satisfy [ψ (z) , ψ † (z )] = δ (z − z ). The particles in each tube are subject to a local potential due to the optical lattice and to an axial confinement potential, i.e., V (z) = V 0 sin 2 2π λ z + 1 2 mω 2 z z 2 . The optical lattice is characterized by its depth V 0 and wavelength λ. The axial confinement is characterized by the frequency ω z and the mass m of the Rb 87 atoms. It is further assumed that the particles interact via a short-range potential U (z − z ) = g 1D δ (z − z ). 32 The chemical potential µ controls the total number of particles in the tube. In order to study the Hamiltonian in Eq. (1), one can then map the problem onto a simplified one-band Bose-Hubbard model, 21 which is a good approximation when the lattice is deep enough so that multiband effects can be neglected. Alternatively, it is possible to account for such effects, that are potentially relevant for the lowest lattice depths considered in the experiments, by introducing a grid spacing = λ/T to obtain the following "artificial" lattice Hamiltonian 24 where we have identified b † i = ψ † (i ) / √ . Here, the hopping matrix element t h = E rec potential (which is characterized by its frequency ω xy ) to the overall chemical potential of the system. The latter is, in turn, fixed by total number of particles in the experiments. We simulate arrays of up to 110 × 110 tubes, each tube being 1000 long. The axial size of the system is chosen so that the expanding particles never reach the edges of the tubes in the time scales of the simulation. The initial state (t ev = 0) is assumed to be the ground state in the absence of tunneling between the 1D tubes. We tune the t ev = 0 axial and transverse trapping potentials (using the experimental value for their ratio) so that the initial fraction of singlons in the simulation matches the experiment. The lattice parameters are those of the experiment. For t ev ≥ 0 the overall confining potential is set to zero exactly. All calculations are carried out using a Gutzwiller meanfield approach detailed in Refs. 16, 23.
The parameters U/J that correspond to the one-band Bose-Hubbard model discussed in the main text were obtained using maximally-localized generalized Wannier states 20 through the software available in Ref. 33.