Quantum many-body systems can have phase transitions1 even at zero temperature; fluctuations arising from Heisenberg’s uncertainty principle, as opposed to thermal effects, drive the system from one phase to another. Typically, during the transition the relative strength of two competing terms in the system’s Hamiltonian changes across a finite critical value. A well-known example is the Mott–Hubbard quantum phase transition from a superfluid to an insulating phase2, 3, which has been observed for weakly interacting bosonic atomic gases. However, for strongly interacting quantum systems confined to lower-dimensional geometry, a novel type4, 5 of quantum phase transition may be induced and driven by an arbitrarily weak perturbation to the Hamiltonian. Here we observe such an effect—the sine–Gordon quantum phase transition from a superfluid Luttinger liquid to a Mott insulator6, 7—in a one-dimensional quantum gas of bosonic caesium atoms with tunable interactions. For sufficiently strong interactions, the transition is induced by adding an arbitrarily weak optical lattice commensurate with the atomic granularity, which leads to immediate pinning of the atoms. We map out the phase diagram and find that our measurements in the strongly interacting regime agree well with a quantum field description based on the exactly solvable sine–Gordon model8. We trace the phase boundary all the way to the weakly interacting regime, where we find good agreement with the predictions of the one-dimensional Bose–Hubbard model. Our results open up the experimental study of quantum phase transitions, criticality and transport phenomena beyond Hubbard-type models in the context of ultracold gases.
Figures at a glance
Figure 1: Comparing two types of superfluid-to-Mott-insulator phase transition in one dimension. 
Schematic density distributions (grey) in the presence of a periodic potential (red solid line). a, Mott–Hubbard-type quantum phase transition for weak interactions3. The system is still superfluid at finite lattice depth (top). The transition to the insulating state is induced by increasing the lattice depth above a finite critical value (bottom). b, Sine–Gordon-type quantum phase transition for strong interactions6. In the absence of any perturbation, the system is a strongly correlated superfluid (top). For sufficiently strong interactions, not necessarily infinitely strong, an arbitrarily weak perturbation by a lattice potential commensurate with the system’s granularity induces the transition to the insulating Mott state (bottom).
Figure 2: Modulation spectroscopy on bosons in one dimension. 
a, b, d, Excitation spectra for low (a), intermediate (b), and high (d) lattice depth, V. The change, δ, in the spatial width after amplitude modulation is plotted as a function of the modulation frequency, f, for different values of γ. a, Characteristic spectra for V = 1.5(1)ER in the superfluid (squares: a3D = 115(2)a0, γ = 1.0(1)) and in the Mott regime (circles: a3D = 261(2)a0, γ = 3.1(2)). The solid lines are linear fits to the high-frequency parts of the spectra. We determine the axis intercepts, fg, as indicated. b, Spectra for V = 3.0(2)ER. The system is superfluid for γ = 0.51(6) (squares) and exhibits a gap for γ = 1.6(1) (triangles) and γ = 4.1(3) (circles). c, Determination of the transition point for the case of the shallow lattice with V = 1.5(1)ER. The frequency fg is plotted as a function of γ. The solid line is an error-function fit to the data. Inset, fg as a function of γ for V = 3.0(2)ER. d, Spectra for V = 9.0(5)ER for weak (squares: γ = 0.10(3)) and strong (circles: γ = 8.1(4)) interactions in the superfluid (SF) and Mott-insulator (MI) regimes. Here we plot f in units of U/h. Modulation parameters and error bars are discussed in Methods.
Figure 3: Transport measurements on the 1D Bose gas. 
Centre-of-mass displacement, x0, as a function of a3D for V = 9.0(5)ER (diamonds), V = 5.0(3)ER (squares) and V = 2.0(1)ER (circles). We extrapolate the linear slope at small values of a3D and associate the transition point with the axis intercept. For the data with V = 2.0(1)ER, transport is not fully quenched as the condition of commensurability is not fulfilled for all atoms. All errors are the 1σ statistical error. Inset, the measured critical ratio (U/J)c at the transition point as a function of lattice depth, V. The dashed line indicates the theoretical result, (U/J)c ≈ 3.85, in the 1D Bose–Hubbard regime26.
Figure 4: Phase diagram for the strongly interacting 1D Bose gas. 
The plane of inverse Lieb–Liniger interaction parameter, 1/γ, and optical lattice depth, V (in units of ER), showing the superfluid and Mott-insulating phases in one dimension. The critical interaction parameter is γc. For strong interactions and shallow lattices, we determine the transition by amplitude modulation spectroscopy (circles). For weak interactions and deep lattices, we probe the phase boundary using transport measurements (squares). The solid and dashed lines are the predictions from the sine–Gordon and Bose–Hubbard models, respectively. Error bars are discussed in the Methods. Inset, the measured gap energy, Eg = hfg, as a function of V for γ = 11(1), and comparison between our data and the analytical result for finite γ as given by the sine–Gordon model (solid line; see Methods). Also shown is the universal behaviour, Eg = V/2, which is valid for non-interacting fermions (dashed line).
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