The ‘Higgs’ amplitude mode at the two-dimensional superfluid/Mott insulator transition
(26 July 2012)
25 April 2012
16 May 2012
25 July 2012
Spontaneous symmetry breaking plays a key role in our understanding of nature. In relativistic quantum field theory, a broken continuous symmetry leads to the emergence of two types of fundamental excitation: massless Nambu–Goldstone modes and a massive ‘Higgs’ amplitude mode. An excitation of Higgs type is of crucial importance in the standard model of elementary particle physics
, and also appears as a fundamental collective mode in quantum many-body systems 1 . Whether such a mode exists in low-dimensional systems as a resonance-like feature, or whether it becomes overdamped through coupling to Nambu–Goldstone modes, has been a subject of debate 2 . Here we experimentally find and study a Higgs mode in a two-dimensional neutral superfluid close to a quantum phase transition to a Mott insulating phase. We unambiguously identify the mode by observing the expected reduction in frequency of the onset of spectral response when approaching the transition point. In this regime, our system is described by an effective relativistic field theory with a two-component quantum field 2 , 3 , 4 , 5 , 6 , 7 , 8 , 9 , which constitutes a minimal model for spontaneous breaking of a continuous symmetry. Additionally, all microscopic parameters of our system are known from first principles and the resolution of our measurement allows us to detect excited states of the many-body system at the level of individual quasiparticles. This allows for an in-depth study of Higgs excitations that also addresses the consequences of the reduced dimensionality and confinement of the system. Our work constitutes a step towards exploring emergent relativistic models with ultracold atomic gases. 2 , 7
Figures at a glance
Figure 1: Illustration of the Higgs mode and experimental sequence.
a, Classical energy density V as a function of the order parameter . Within the ordered (superfluid) phase, Nambu–Goldstone and Higgs modes arise from phase and amplitude modulations (blue and red arrows in panel 1). As the coupling Ψ j = J/ U (see main text) approaches the critical value j , the energy density transforms into a function with a minimum at c = 0 (panels 2 and 3). Simultaneously, the curvature in the radial direction decreases, leading to a characteristic reduction of the excitation frequency for the Higgs mode. In the disordered (Mott insulating) phase, two gapped modes exist, respectively corresponding to particle and hole excitations in our case (red and blue arrow in panel 3). Ψ b, The Higgs mode can be excited with a periodic modulation of the coupling j, which amounts to a ‘shaking’ of the classical energy density potential. In the experimental sequence, this is realized by a modulation of the optical lattice potential (see main text for details). τ = 1/ ν ; mod E , lattice recoil energy. r
Figure 2: Softening of the Higgs mode.
a, The fitted gap values h ν / 0 U (circles) show a characteristic softening close to the critical point in quantitative agreement with analytic predictions for the Higgs and the Mott gap (solid line and dashed line, respectively; see text). Horizontal and vertical error bars denote the experimental uncertainty of the lattice depths and the fit error for the centre frequency of the error function, respectively (Methods). Vertical dashed lines denote the widths of the fitted error function and characterize the sharpness of the spectral onset. The blue shading highlights the superfluid region. b, Temperature response to lattice modulation (circles and connecting blue line) and fit with an error function (solid black line) for the three different points labelled in a. As the coupling j approaches the critical value j , the change in the gap values to lower frequencies is clearly visible (from panel 1 to panel 3). Vertical dashed lines mark the frequency c U/ h corresponding to the on-site interaction. Each data point results from an average of the temperatures over ~50 experimental runs. Error bars, s.e.m.
Figure 3: Theory of in-trap response.
a, A diagonalization of the trapped system in a Gutzwiller approximation shows a discrete spectrum of amplitude-like eigenmodes. Shown on the vertical axis is the strength of the response to a modulation of j. Eigenmodes of phase type are not shown (Methods) and ν denotes the gap as calculated in the Gutzwiller approximation. a.u., arbitrary units. 0,G b, In-trap superfluid density distribution for the four amplitude modes with the lowest frequencies, as labelled in a. In contrast to the superfluid density, the total density of the system stays almost constant (not shown). c, Discrete amplitude mode spectrum for various couplings j/ j . Each red circle corresponds to a single eigenmode, with the intensity of the colour being proportional to the line strength. The gap frequency of the lowest-lying mode follows the prediction for commensurate filling (solid line; same as in c Fig. 2a) until a rounding off takes place close to the critical point due to the finite size of the system. d, Comparison of the experimental response at V = 9.5 0 E (blue circles and connecting blue line; error bars, s.e.m.) with a 2 r × 2 cluster mean-field simulation (grey line and shaded area) and a heuristic model (dashed line; for details see text and Methods). The simulation was done for V = 9.5 0 E (grey line) and for r V = (1 0 ± 0.02) × 9.5 E (shaded grey area), to account for the experimental uncertainty in the lattice depth, and predicts the energy absorption per particle r Δ E.
Figure 4: Scaling of the low-frequency response.
The low-frequency response in the superfluid regime shows a scaling compatible with the prediction (1
− j/ j ) c −2 ν (Methods). Shown is the temperature response rescaled with (1 3 − j/ j ) c for 2 V = 10 0 E (grey), 9.5 r E (black), 9 r E (green), 8.5 r E (blue) and 8 r E (red) as a function of the modulation frequency. The black line is a fit of the form r a with a fitted exponent ν b b = 2.9(5). The inset shows the same data points without rescaling, for comparison. Error bars, s.e.m.
Figure 5: Response between the strongly interacting limit and the weakly interacting limit.
a, Change in temperature Δ T as a function of j/ j and the modulation frequency c ν in units of mod U. A pronounced feature close to j/ j = 1 directly shows the existence of the gap and its softening, which is also observed in units of c J ( Supplementary Fig. 8). As the weakly interacting limit (higher values of j/ j ) is approached, the response broadens and vanishes. c b, Simulation using a variational 2 × 2 cluster wavefunction predicting the energy absorption per particle Δ E for the same parameter range. The simulation shows agreement with the experimental data near the critical point in both the softening of the response and the overall width of the absorption band. However, the simulation does not fully reproduce the vanishing of the response at higher j/ j values. A splitting in the excitation structure at c j/ j c ≈ 3 is visible, which might also be present in the experimental data. A low-frequency feature associated with density oscillations at the edges of the trap due to the excitation of phase-like modes is clearly seen in the simulations. This feature occurs below the lowest measured frequency in the experiment and thus is not visible in a, (except in the vicinity of the critical point, where the lowest modulation frequencies are close to this feature). Black solid lines show the mean-field predictions as plotted in Fig. 2a.