Interactions lie at the heart of correlated many-body quantum phases1, 2, 3. Typically, the interactions between microscopic particles are described as two-body interactions. However, it has been shown that higher-order multi-body interactions could give rise to novel quantum phases with intriguing properties. So far, multi-body interactions have been observed as inelastic loss resonances in three- and four-body recombinations of atom–atom and atom–molecule collisions4, 5, 6. Here we demonstrate the presence of effective multi-body interactions7 in a system of ultracold bosonic atoms in a three-dimensional optical lattice, emerging through virtual transitions of particles from the lowest energy band to higher energy bands. We observe such interactions up to the six-body case in time-resolved traces of quantum phase revivals8, 9, 10, 11, using an atom interferometric technique that allows us to precisely measure the absolute energies of atom number states at a lattice site. In addition, we show that the spectral content of these time traces can reveal the atom number statistics at a lattice site, similar to foundational experiments in cavity quantum electrodynamics that yield the statistics of a cavity photon field12. Our precision measurement of multi-body interaction energies provides crucial input for the comparison of optical-lattice quantum simulators with many-body quantum theory.
Figures at a glance
Figure 1: Signature of multi-body interactions in quantum phase revivals. 
a, A BEC loaded into a weak optical lattice forms a superfluid in which each atom is delocalized over several lattice sites. The quantum state at each site can be expressed as a superposition of Fock states, |n
, with amplitudes cn. The number of blue balls in each state indicates the number of atoms, n. b, For repulsive interactions, virtual transitions to higher lattice orbitals broaden the ground-state wavefunction at a lattice site depending on the atom number (orange solid lines) relative to the wavefunction in a non-interacting system (grey dashed lines). This gives rise to characteristic Fock state energies, which can be described by effective multi-body interactions. c, Quantum phase revivals of a coherent state of interacting atoms in the multi-orbital system of a deep lattice well (blue solid line). The beat signal indicates coherent multi-body interactions. The dynamics are markedly different from the monochromatic evolution expected in a single-orbital model with a single two-body interaction energy, U (grey solid line).
Figure 2: Multi-orbital quantum phase revivals of atom number superposition states. 
a, Collapse and revival dynamics of number-squeezed superposition states in a deep optical lattice. A BEC of about (1.9 ± 0.3) × 105 87Rb atoms was adiabatically loaded into a VL = 8Erec lattice within 100 ms. Quantum phase evolution was induced by a non-adiabatic jump to a deep, VH = (41.0 ± 1.3)Erec, lattice, preserving superposition states with finite number fluctuations, an ensemble-averaged mean atom number of
≈ 1.0 and a central mean atom number of 
≈ 2.5. Simultaneously with the lattice jump, the underlying harmonic confinement was instantaneously minimized to optimize the coherence time. The quantum phase dynamics show a beat-note signature resulting from coherent multi-body interactions. Each data point corresponds to a single run of the experiment. The solid line interpolates the data and serves as a guide to the eye. b, Spectral analysis of the time trace (a) reveals the contributing frequencies. The solid line shows Gaussian fits to the peaks. Grey dashed lines display the frequencies corresponding to the single-orbital interaction energies U and 2U at a lattice depth of 41Erec. a.u., arbitrary units.
Figure 3: Multi-orbital energies and effective multi-body interactions. 
a, Long collapse and revival traces were recorded under identical loading conditions (VL = 8Erec) but variable lattice depths, VH, during phase evolution. Numerical Fourier transforms of the time traces reveal the contributing frequencies of orders U/h (red circles) and 2U/h (blue circles), which have a typical uncertainty of ±50 Hz. The shading of the data points reflects the relative spectral weight (lighter, lower; darker, higher). The solid lines with grey shading indicate the theoretically expected frequencies at an s-wave scattering length of as = (102 ± 2)a0 (a0, Bohr radius), as derived for a basis set with 43 orbitals. Calculation using a smaller basis set, with 33 orbitals, yields slightly higher energies (dashed lines). The black dotted lines show the single-orbital interaction energies U and 2U of the Bose–Hubbard model. At low lattice depths, only the strongest peaks can be resolved, mainly as a result of smaller peak spacings. b, Effective two-body (top), three-body (middle) and four-body (bottom) interaction strengths as derived from experiment and theory (as = (102 ± 2)a0, 43 orbitals). Error bars, 1 s.d.
Figure 4: Global number statistics on approaching the Mott insulator transition. 
a, Multi-orbital quantum phase revivals in a deep, VH = 40Erec, lattice after adiabatic loading of (3.3 ± 0.3) × 105 87Rb atoms into lattices with depths ranging from VL = 3Erec to 13Erec. The mean atom number of the individual traces differed by as little as ±1%. Although the coherence time in shallow lattices is significantly reduced, the visibility reliably shows dynamics down to the per cent level (inset). b, The corresponding Fourier spectra reveal frequency contributions from Fock states containing up to six atoms. The peak positions agree with the theoretical predictions (dashed vertical lines) and are independent of VL. Number squeezing manifests itself both in reduced peak amplitudes and in a narrowing of the spectra for increasing VL. The solid lines show Gaussian fits to the peaks.
