Microscopic mechanisms of phase-coherence decay in a two-dimensional optical lattice of tunnel-coupled one-dimensional Bose gases

When a Bose–Einstein condensate is stored in a two-dimensional optical lattice, it is split into thousands of narrow potential tubes, each holding a one-dimensional quantum gas, with the tunnel coupling between neighboring tubes controlled by the lattice depth. In the original 2001 experiment of Greiner, Bloch, Mandel, Hänsch and Esslinger, the phase coherence of this tube array was strikingly long lived, yet the measured condensate fraction decayed with hold time, faster at larger lattice depths; the authors conjectured — but did not prove — that this decay is dephasing driven by residual lattice-potential fluctuations amplified by the reduced tunnel coupling, and they could not determine whether atoms inside individual tubes remain in the axial ground state or carry axial excitations. The open problem is to quantitatively identify the dominant microscopic mechanisms (technical lattice noise versus quantum phase diffusion versus other heating and loss channels) behind the coherence and condensate-fraction decay of tunnel-coupled one-dimensional Bose gases in a two-dimensional optical lattice, including the noise-to-band-dynamics mapping and the resolution of the axial excitation content of individual tubes, while the strongly interacting (Tonks–Girardeau) regime and the three-dimensional superfluid–Mott-insulator transition envisaged in the original context have since been realized.

Background

An optical lattice is a periodic potential for neutral atoms created by the dipole force of interfering laser beams; its depth is conventionally measured in units of the recoil energy $E_r=\hbar^2k^2/2m$, the kinetic energy an atom gains by absorbing one lattice photon of wave vector $k$. Greiner, Bloch, Mandel, Hänsch and Esslinger (Phys. Rev. Lett. 87, 160405 (2001)) overlapped two perpendicular standing-wave laser fields on a rubidium Bose–Einstein condensate, producing a two-dimensional lattice of tightly confining tubes each filled with a one-dimensional (1D) quantum gas — up to 3000 tubes with about 170 atoms each. In such tubes the radial motion is frozen to zero-point oscillations (trap frequency about 2π × 18.5 kHz at a depth of 12 $E_r$), while the axial motion along the tube is only weakly trapped (2π × 10–300 Hz). The tunnel coupling between neighboring tubes — the matrix element $J$ with which an atom moves between adjacent lattice sites — decreases exponentially with lattice depth $U_0$, so the array spans regimes from a single coherent condensate to nearly isolated tubes. When all potentials are switched off suddenly, the expanding matter waves from the thousands of tubes interfere, and the resulting diffraction peaks are a direct image of the phase coherence of the condensate wave function across the lattice: a sharp interference pattern requires a well-defined relative phase between neighboring tubes.

The experiment found this coherence to be remarkably long lived, but measured the number of condensed atoms after a variable hold time and observed a lifetime reduction that grows with lattice depth. The authors' proposed explanation was deliberately conjectural: residual fluctuations of the lattice potential dephase neighboring condensates, and this dephasing is faster at larger $U_0$ because the tunneling matrix element, which could re-lock the phases, is then strongly reduced. In a band-structure picture — the energy bands of the periodic potential, with momenta folded into the first Brillouin zone, the elementary cell of reciprocal space — the bands narrow strongly at large depth, so external perturbations more easily cause nonadiabatic transitions within a single band, leaving atoms in each site's ground state without phase coherence to their neighbors. Consistent with this, a band-mapping measurement (adiabatically ramping the lattice down so that quasimomentum maps to real momentum) showed that even after long storage, when no interference pattern and no measurable condensate fraction remain, all atoms still populate the first energy band and hence the radial ground state of a lattice site. Two limitations were explicitly acknowledged: the causal chain from lattice fluctuations to the observed decay was unproven and no quantitative characterization of the noise sources and their effect on band dynamics existed; and the measurement could not distinguish whether the atoms in a single tube also remain in the axial ground state of the tube, or whether axial (along-tube) excitations are present — band mapping resolves the radial and lattice-band, but not the axial, degree of freedom.

The experiment was part of a broader program. Jaksch, Bruder, Cirac, Gardiner and Zoller (Phys. Rev. Lett. 81, 3108 (1998)) had predicted that bosons in a three-dimensional optical lattice undergo a quantum phase transition from a superfluid to a Mott-insulator state (a state with a fixed integer number of atoms per site and suppressed phase coherence), which the same Munich group realized shortly afterwards (Greiner et al., Nature 415, 39 (2002)). The 2001 paper also closed with the outlook that lowering the density per tube should give access to quasi-condensates (one-dimensional gases with strong thermal density fluctuations but a phase that remains smooth over long distances) and to the Tonks–Girardeau gas, the 1D limit in which bosons become impenetrable and map onto free fermions, which was subsequently achieved in lattice-tube arrays (Paredes et al., Nature 429, 277 (2004); Stöferle et al., Phys. Rev. Lett. 92, 130403 (2004)) and in free 1D geometry (Kinoshita, Wenger and Weiss, Science 305, 1125 (2004)).

In parallel, the general theory of the mechanisms at stake matured. Quantum phase diffusion — the growth of random relative phases between sites caused by number fluctuations at each site, the discrete analog of phase diffusion in split condensates — was analyzed for lattice condensates by Xiong, Liu, Huang and Wang (J. Phys. B 36, 3315 (2003)), and the interference patterns of condensates in a two-dimensional lattice were computed by Liu, Xiong, Xu and Huang (J. Phys. B 36, 2083 (2003)). The effect of technical noise — fluctuations of the lattice laser intensity, phase and pointing — was quantified for single atoms by Gerbier and Castin (Phys. Rev. A 82, 013615 (2010)), who derived heating rates including a ground-band heating rate that measures energy growth within the lowest band, and for interacting bosons by Pichler, Schachenmayer, Daley and Zoller (Phys. Rev. A 87, 033606 (2013)), who derived a master equation (an evolution equation for the quantum state's density matrix, with the noise acting as an environment) for low-frequency intensity noise and computed heating rates and correlation decay for many-body states. Later experiments with one-dimensional gases probed the axial degree of freedom directly in other geometries: the quantum Newton's cradle of Kinoshita, Wenger and Weiss (Nature 440, 900 (2006)) showed that axial dynamics in 1D gases fails to thermalize, and Hofferberth et al. (Nature 449, 324 (2007)) studied the non-equilibrium coherence dynamics of split 1D Bose gases. What none of this established is a definitive, quantitative attribution of the decay observed in the two-dimensional tube-array lattice, or a resolution of the axial ground-state question for stored lattice tubes.

Problem Statement

For a degenerate Bose gas stored in a two-dimensional optical lattice that forms an array of tunnel-coupled one-dimensional tubes — the system of Greiner, Bloch, Mandel, Hänsch and Esslinger, Phys. Rev. Lett. 87, 160405 (2001) — two linked questions remain unresolved, and together they define the problem:

1. Identify, with quantitative evidence, the dominant microscopic mechanism(s) responsible for the observed reduction of phase coherence and condensate fraction with hold time in the lattice, a reduction that accelerates as the lattice depth $U_0$ increases and the tunnel coupling between tubes decreases. Identifying the dominant mechanism requires (a) characterizing the residual lattice-potential noise — intensity, phase and pointing fluctuations of the lattice beams — at the frequencies relevant to the band dynamics; (b) computing the effect of this noise on the band populations and phase coherence, including the nonadiabatic intra-band dynamics that the source's band-narrowing argument invokes; (c) discriminating quantitatively between dephasing driven by residual lattice-potential fluctuations, intrinsic quantum phase diffusion arising from number fluctuations at each site, and any other heating or atom-loss channels, accounting for the fact that dephased condensates that do not recombine on adiabatic release also reduce the measured condensate fraction; and (d) reproducing the measured dependence of the decay on lattice depth and tunnel coupling across the range in which it was observed, consistent with the band-mapping constraint that atoms remain in the lowest transverse band and in the first Bloch band of the lattice throughout.

2. Determine whether the atoms within an individual lattice tube remain in the axial (along-tube) ground state after storage and dephasing, or whether axial excitations are present — the question the original band-mapping measurement explicitly could not resolve, since it constrains the radial and lattice-band degrees of freedom but not the weakly confined axis of a single tube.

The strongly interacting regimes of the 1D gases (quasi-condensates, Tonks–Girardeau gas) and the extension to three-dimensional lattices with the superfluid–Mott-insulator transition, which the original context envisaged, are treated as resolved by later landmark experiments and are not part of this problem.

References

1. M. Greiner, I. Bloch, O. Mandel, T. W. Hänsch, T. Esslinger, 'Exploring Phase Coherence in a 2D Lattice of Bose-Einstein Condensates', Phys. Rev. Lett. 87, 160405 (2001). DOI: 10.1103/PhysRevLett.87.160405, https://doi.org/10.1103/PhysRevLett.87.160405
2. D. Jaksch, C. Bruder, J. I. Cirac, C. W. Gardiner, P. Zoller, 'Cold Bosonic Atoms in Optical Lattices', Phys. Rev. Lett. 81, 3108 (1998). DOI: 10.1103/PhysRevLett.81.3108, https://doi.org/10.1103/PhysRevLett.81.3108
3. M. Greiner, O. Mandel, T. Esslinger, T. W. Hänsch, I. Bloch, 'Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms', Nature 415, 39 (2002). DOI: 10.1038/415039a, https://doi.org/10.1038/415039a
4. P. Öhberg, L. Santos, 'Dynamical Transition from a Quasi-One-Dimensional Bose-Einstein Condensate to a Tonks-Girardeau Gas', Phys. Rev. Lett. 89, 240402 (2002). DOI: 10.1103/PhysRevLett.89.240402, https://doi.org/10.1103/PhysRevLett.89.240402
5. H. Xiong, S. Liu, G. Huang, L. Wang, 'Phase diffusion of Bose-Einstein condensates in a one-dimensional optical lattice', J. Phys. B: At. Mol. Opt. Phys. 36, 3315 (2003). DOI: 10.1088/0953-4075/36/15/312, https://doi.org/10.1088/0953-4075/36/15/312
6. S. Liu, H. Xiong, Z. Xu, G. Huang, 'Interference patterns of Bose-condensed gases in a two-dimensional optical lattice', J. Phys. B: At. Mol. Opt. Phys. 36, 2083 (2003). DOI: 10.1088/0953-4075/36/10/316, https://doi.org/10.1088/0953-4075/36/10/316
7. B. Paredes, A. Widera, V. Murg, O. Mandel, S. Fölling, I. Cirac, G. V. Shlyapnikov, T. W. Hänsch, I. Bloch, 'Tonks–Girardeau gas of ultracold atoms in an optical lattice', Nature 429, 277 (2004). DOI: 10.1038/nature02530, https://doi.org/10.1038/nature02530
8. T. Kinoshita, T. Wenger, D. S. Weiss, 'Observation of a One-Dimensional Tonks-Girardeau Gas', Science 305, 1125 (2004). DOI: 10.1126/science.1100700, https://doi.org/10.1126/science.1100700
9. T. Stöferle, H. Moritz, C. Schori, M. Köhl, T. Esslinger, 'Transition from a Strongly Interacting 1D Superfluid to a Mott Insulator', Phys. Rev. Lett. 92, 130403 (2004). DOI: 10.1103/PhysRevLett.92.130403, https://doi.org/10.1103/PhysRevLett.92.130403
10. F. Gerbier, A. Widera, S. Fölling, O. Mandel, T. Gericke, I. Bloch, 'Phase Coherence of an Atomic Mott Insulator', Phys. Rev. Lett. 95, 050404 (2005). DOI: 10.1103/PhysRevLett.95.050404, https://doi.org/10.1103/PhysRevLett.95.050404
11. T. Kinoshita, T. Wenger, D. S. Weiss, 'A quantum Newton's cradle', Nature 440, 900 (2006). DOI: 10.1038/nature04693, https://doi.org/10.1038/nature04693
12. S. Hofferberth, I. Lesanovsky, B. Fischer, T. Schumm, J. Schmiedmayer, 'Non-equilibrium coherence dynamics in one-dimensional Bose gases', Nature 449, 324 (2007). DOI: 10.1038/nature06149, https://doi.org/10.1038/nature06149
13. F. Gerbier, Y. Castin, 'Heating rates for an atom in a far-detuned optical lattice', Phys. Rev. A 82, 013615 (2010). DOI: 10.1103/PhysRevA.82.013615, https://doi.org/10.1103/PhysRevA.82.013615 (arXiv:1002.5018)
14. H. Pichler, J. Schachenmayer, A. J. Daley, P. Zoller, 'Heating dynamics of bosonic atoms in a noisy optical lattice', Phys. Rev. A 87, 033606 (2013). DOI: 10.1103/PhysRevA.87.033606, https://doi.org/10.1103/PhysRevA.87.033606 (arXiv:1301.2803)
15. R. Vatré, R. Bouganne, M. Bosch Aguilera, A. Ghermaoui, J. Beugnon, R. Lopes, F. Gerbier, 'Dynamics of spatial phase coherence in a dissipative Bose–Hubbard atomic system', Comptes Rendus. Physique 24, 263 (2024). DOI: 10.5802/crphys.166, https://doi.org/10.5802/crphys.166
16. R. Vatré, G. Morettini, J. Beugnon, R. Lopes, L. Mazza, F. Gerbier, 'Phase Coherence of Strongly Interacting Bosons in One-Dimensional Optical Lattices', arXiv:2603.26118 (2026). arXiv:2603.26118, https://arxiv.org/abs/2603.26118