Measuring the dynamic structure factor of a quantum gas undergoing a structural phase transition

Abstract

The dynamic structure factor is a central quantity describing the physics of quantum many-body systems, capturing structure and collective excitations of a material. In condensed matter, it can be measured via inelastic neutron scattering, which is an energy-resolving probe for the density fluctuations. In ultracold atoms, a similar approach could so far not be applied because of the diluteness of the system. Here we report on a direct, real-time and nondestructive measurement of the dynamic structure factor of a quantum gas exhibiting cavity-mediated long-range interactions. The technique relies on inelastic scattering of photons, stimulated by the enhanced vacuum field inside a high finesse optical cavity. We extract the density fluctuations, their energy and lifetime while the system undergoes a structural phase transition. We observe an occupation of the relevant quasi-particle mode on the level of a few excitations, and provide a theoretical description of this dissipative quantum many-body system.

Figure 1. Scheme for measuring the dynamic structure factor in a quantum gas.

(a) An incident laser beam (red) with wave vector k_i is spontaneously scattered at an atomic cloud (blue) into a free-space mode with wave vector k_f. Analysing the scattered photons as a function of their frequency shift ω_f−ω_i and magnitude yields the dynamic structure factor. The resulting signal for dilute quantum gases is vanishingly small, as indicated by the small size of the arrow pointing towards the detector. (b) Atoms placed into an optical high-finesse resonator (light blue) feel a strongly enhanced vacuum field (golden). Their spontaneous scattering rate into the mode k_f can hereby be increased by orders of magnitude, resulting in a detectable signal for the dynamic structure factor.

Figure 2. Power spectral density and dynamic structure factor.

(a) The power spectral density PSD of the light field leaking out of the cavity is shown as a function of frequency shift ω with respect to ω_p and relative transverse pump power P/P_cr (average over 147 experimental repetitions). Two sidebands are visible, corresponding to the incoherent creation (ω<0) and annihilation (ω>0) of quasi-particles. The energy of these quasi-particles vanishes towards the critical point. At the phase transition, a strong coherent field at the pump frequency appears (ω=0). We attribute the broadened feature around ω=0 to residual low-frequency technical noise in our system. Note the large dynamic range of the data on the logarithmic scale. The panels (b–d) show the normalized dynamic structure factor S(k_cb, ω) for three different values of P/P_cr (see dashed lines in upper panel), derived from the power spectral density PSD(ω). While the position and width of the sidebands give direct access to the energy and lifetime of the quasi-particles, the sideband asymmetry can be used to determine the occupation of the quasi-particle mode. Red line shows a fit to the sidebands with our theoretical model to extract these properties.

Figure 3. Density fluctuations and density modulation.

The variance of density fluctuations (filled blue symbols) and the square of the density modulation (open black symbols) of the long-range interacting quantum gas is shown as a function of relative pump power P/P_cr. The fluctuation data are extracted from the dynamic structure factor (see Fig. 2) by integrating over the fit function describing the sidebands and is proportional to the static structure factor. The coherent density modulation is calculated from the power spectral density at the zero frequency bin. The vertical error bars display the statistical error (s. d.) from the fit, while the horizontal error bars display the s.d. in our determination of the critical point. The inset displays a double logarithmic plot to demonstrate the scaling behaviour of the variance of the density fluctuations against the distance to the critical point, expressed as the Hamiltonian coupling parameter λ (see Supplementary Note 1). From a linear fit, we find critical exponents of 0.7(1) and 1.1(1) on the normal and self-organized side, respectively. The open symbols in the inset are used for the fitting.

Figure 4. Characterization of the quasi-particle mode.

Frequency ω_s (a) and decay rate γ (b) of the quasi-particles as a function of relative pump power P/P_cr, as extracted from the fit to the dynamic structure factor (Fig. 2). The grey-shaded area in the top panel results from an ab initio calculation of the expected soft mode frequency, taking into account the experimental uncertainties in the determination of the coherent cavity field and the depth of the optical lattice resulting from the transverse pump field. Close to P/P_cr=1, for ω_s/(2π)<400 Hz, the uncertainty in modelling atom loss leads to a substructure, which we omit in the graph. The solid line in the lower panel is a fit with a phenomenological function to the data (see Supplementary Note 3). Vertical and horizontal error bars indicate the statistical errors (s. d.) reported from the fit, and the error (s.d.) in the determination of the critical point, respectively.

Figure 5. Number of quasi-particles.

Number of quasi-particles as a function of relative pump power P/P_cr, extracted from the sideband asymmetry in the dynamic structure factor. The grey-shaded area shows the result from an ab initio calculation of the expected quasi-particle mode occupation (Supplementary Notes 1 and 3), taking into account the experimental uncertainties in the determination of the coherent cavity field and the depth of the optical lattice resulting from the transverse pump field. Shown as black dashed line is the calculated thermal occupation of the quasi-particle mode because of the finite temperature of the BEC for a temperature of 38 nK. Vertical and horizontal error bars indicate the statistical error (s.d.) reported from the fit, and the error in the determination of the critical point (s.d.), respectively. The strongly increased vertical error bars close to P/P_cr=1 arise from the decreasing sideband asymmetry, while their individual errors stay roughly constant.