Hanbury Brown and Twiss Bunching of Phonons and of the Quantum Depletion in an Interacting Bose Gas

We report the realization of a Hanbury Brown and Twiss (HBT)-like experiment with a gas of interacting bosons at low temperatures. The low-temperature regime is reached in a three-dimensional optical lattice and atom-atom correlations are extracted from the detection of individual metastable helium atoms after a long free fall. We observe, in the noncondensed fraction of the gas, a HBT bunching whose properties strongly deviate from the HBT signals expected for noninteracting bosons. In addition, we show that the measured correlations reflect the peculiar quantum statistics of atoms belonging to the quantum depletion and of the Bogoliubov phonons, i.e., of collective excitations of the many-body quantum state. Our results demonstrate that atom-atom correlations provide information about the quantum state of interacting particles, extending the interest of HBT-like experiments beyond the case of noninteracting particles.

Figure 1

(a) Interacting metastable helium ${}^4{\mathrm{He}}^{*}$ atoms are loaded into a 3D optical lattice of spacing $d=775\mathrm{nm}$. The tunneling energy is denoted $J$ and the on-site interaction energy $U$. In this work, we set $U/J=10$. (b) Sketch of the detection method. Atoms are released from the lattice and reach the ${\mathrm{He}}^{*}$ detector after a long free fall (325 ms) that maps the momentum distribution of the trapped atoms into the measured spatial distribution. The use of a microchannel plate, in combination with delay-line anodes (not shown), allows for the detection of individual atoms in three dimensions [20] from which the HBT correlations are extracted. (c) Dispersion relation of the Bogoliubov excitations of the interacting gas in the lattice (solid blue line, $U/J=10$) and of noninteracting particles in the lattice (dashed orange line, $U/J=0$). The collective nature of the excitations is dominant in the linear part of the Bogoliubov spectrum at low energies, which is associated with phonons.

Figure 2

Two-body HBT correlations in a strongly interacting lattice Bose gas at $T=2.9J$. The plots are 1D cuts through the 3D correlation function along the lattice axis ${\mathbf{u}}_x$ with $\mathrm{\Delta }{k}_{\perp }={10}^{-2}{k}_d$. (a) Two-body correlation function $g_{{\mathrm{\Omega }}_{\mathbf{k}}}^{(2)}(\delta k)$ in the condensate. Inset: the red sphere depicts the volume ${\mathrm{\Omega }}_{\mathbf{k}}$ ($|\mathbf{k}|<0.04k_d$) over which the correlations are calculated. We find $g_{{\mathrm{\Omega }}_{\mathbf{k}}}^{(2)}(\delta k_x)=1.0(1)$ for the condensate mode, i.e., no bunching as expected when one mode only is populated. (b) Two-body correlation function $g_{{\mathrm{\Omega }}_{\mathbf{k}}}^{(2)}(\delta k)$ in the noncondensed fraction. Inset: the green region depicts the volume ${\mathrm{\Omega }}_{\mathbf{k}}$ ($0.1k_d<|\mathbf{k}|<0.5k_d$) over which the correlations are calculated. Note that the peaks associated with the condensate are excluded from this volume. One observes a well contrasted bunching whose bell shape is fitted with a Gaussian function (dashed line) to quantify the bunching properties.

Figure 3

(a) Bunching amplitude $g^{(2)}(0)-1$ in the noncondensed fraction as a function of the reduced temperature $k_{B}T/\mu$. The measurements are consistent with $g^{(2)}(0)=2$, i.e., with a chaotic statistics at any temperature. The vertical blue dashed line in both panels signals $T_{\mathrm{BEC}}$. (b) Two-body correlation width ${\sigma }_k$ plotted as a function of $k_{B}T/\mu$. For each temperature, the value of the temperature-dependent interaction energy $\mu (T)=n_0(T)U$ in the experiment is calculated from the lattice filling $n_0$ at the trap center. ${\sigma }_k$ is expressed in units of $(R_{\mu }{)}^{-1}=\sqrt{m{\omega }^2/2\mu (T)}$. $R_{\mu }$ coincides with the BEC radius at $T=0$, $R_{\mu }(T=0)=R_{\mathrm{BEC}}$. The red square corresponds to a numerical calculation of the correlation width ${\sigma }_k^B$ for a harmonically trapped one-dimensional interacting Bose gas in the Bogoliubov approximation, with parameters consistent with the three-dimensional experiment (see main text and [33]). The dashed black line is the prediction ${\sigma }_k^{\text{ideal}}=\sqrt{\hslash \omega /k_{B}T}$ for noninteracting bosons at thermal equilibrium, for which ${\sigma }_k^{\text{ideal}}{R}_{\mu }=\sqrt{2\mu /k_{B}T}$.