Direct Observation of Quantum Phonon Fluctuations in a One-Dimensional Bose Gas

We report the first direct observation of collective quantum fluctuations in a continuous field. Shot-to-shot atom number fluctuations in small subvolumes of a weakly interacting, ultracold atomic 1D cloud are studied using in situ absorption imaging and statistical analysis of the density profiles. In the cloud centers, well in the quantum quasicondensate regime, the ratio of chemical potential to thermal energy is $\mu /k_{B}T\simeq 4$, and, owing to high resolution, up to 20% of the microscopically observed fluctuations are quantum phonons. Within a nonlocal analysis at variable observation length, we observe a clear deviation from a classical field prediction, which reveals the emergence of dominant quantum fluctuations at short length scales, as the thermodynamic limit breaks down.

Figure 1

Microscopic fluctuations and thermodynamic theories, at $T=18\mathrm{nK}$ (b) and $T=4.7\mathrm{nK}$ (a),(c). (a) Typical absorption picture (optical density). (b),(c) Normalized fluctuations and average profiles (insets). Theoretical curves: classical shot noise (dotted), predictions from Eq. (1) and the ideal Bose gas (short-dashed), quasicondensate (long-dashed), and Yang-Yang EoS (solid). Gray areas in the profiles indicate the quantum quasicondensate regime, the dark-gray area in the $T=4.7\mathrm{nK}$ profile is the zone $\Omega$ where the analysis of Fig. 4 is carried.

Figure 2

Quantum vs thermal fluctuations in 1D quasicondensates. (a),(b) Fluctuation regimes for thermal (a) and quantum (b) quasicondensates. (c) $S_k^Q$ (solid) and $S_k^T$ (dashed) in Bogoliubov theory for various $k_{B}T/\mu$. (d) $g^{(2)}(z)$ for $\gamma =0.01$.

Figure 3

Quantum vs thermal contributions in the measured fluctuations $⟨\delta N^2{⟩}_m$ (same data as in Fig. 1). Dot-dashed lines are the spectral 1D Bogoliubov predictions $⟨\delta N^2{⟩}_m^Q$ (low-lying green), $⟨\delta N^2{⟩}_m^T$ (middle red) and $⟨\delta N^2{⟩}_m^{\mathrm{tot}}$ (upper blue). Thermodynamic predictions as in Fig. 1 are here rescaled by ${\kappa }_2=0.34$ (a) and 0.41 (b) for $\delta =3.5$ and $2.8\mu \mathrm{m}$, respectively. The square in (b) is the zone $\Omega$.

Figure 4

Fluctuations at variable length scales in the cloud center (zone $\Omega$) at $T=4.7\mathrm{nK}$. (a) Typical profile $N$ (triangles), reference profile $⟨N⟩$ (circles) and fluctuations $\delta N$ (squares), for three pixel sizes $L_{\mathrm{pix}}$. (b) $⟨\delta N^2⟩/⟨N⟩$ in zone $\Omega$ compared to full Bogoliubov (FBM) prediction $⟨\delta N^2{⟩}_m^{\mathrm{tot}}/{\kappa }_2$ (solid), classical field (CFM) prediction $⟨\delta N^2{⟩}_m^T/{\kappa }_2$ (dashed) and thermodynamic G-P prediction (dot-dashed) at $T_{1D}^{*}=5.0\mathrm{nK}$. Inset: same data fitted with FBM at $T_{1D}^{*}=5.0\mathrm{nK}$ (solid) or CFM at $T_{1D}^C=5.5\mathrm{nK}$ (dashed).