Probing the Degree of Coherence through the Full 1D to 3D Crossover

We experimentally study a gas of quantum degenerate ${}^{87}\mathrm{Rb}$ atoms throughout the full dimensional crossover, from a one-dimensional (1D) system exhibiting phase fluctuations consistent with 1D theory to a three-dimensional (3D) phase-coherent system, thereby smoothly interpolating between these distinct, well-understood regimes. Using a hybrid trapping architecture combining an atom chip with a printed circuit board, we continuously adjust the system’s dimensionality over a wide range while measuring the phase fluctuations through the power spectrum of density ripples in time-of-flight expansion. Our measurements confirm that the chemical potential $\mu$ controls the departure of the system from 3D and that the fluctuations are dependent on both $\mu$ and the temperature $T$. Through a rigorous study we quantitatively observe how inside the crossover the dependence on $T$ gradually disappears as the system becomes 3D. Throughout the entire crossover the fluctuations are shown to be determined by the relative occupation of 1D axial collective excitations.

Figure 1

Measured density ripples power spectra (solid blue lines) for a number of traps through the 1D to 3D dimensional crossover, ordered by $\tilde{\mu }=\mu /\hslash {\omega }_{\perp }$ and $\tilde{T}=k_{B}T/\hslash {\omega }_{\perp }$, with trapping frequencies from left to right ${\omega }_{\perp }=2\pi \times (1235,1380,1005,800,740,395)\mathrm{Hz}$ and ${\omega }_z=2\pi \times (16.1,17.4,16.9,20.8,23.2,33.7)\mathrm{Hz}$. For comparison the theoretical predictions are shown, both with (red dash-dotted lines) and without (black dashed lines) the effect of interactions during time of flight. Finite optical imaging resolution has been accounted for by convolution with a Gaussian of width ${\sigma }_{\mathrm{psf}}=4\mathrm{\mu m}$ (see Supplemental Material [36]). Shaded bands represent the statistical uncertainty as 2 standard deviations from the mean, and are obtained via bootstrapping. Insets each depict a representative example of an experimental OD image, showing an individual realization from the corresponding dataset.

Figure 2

Dependence of the amplitude of the phase fluctuations on (a) the aspect ratio of the trapping potential $\kappa$, and (b) the ratio of system length to phase coherence length $\tilde{L}$. (c) Visualisation of the parameter space explored in terms of $\tilde{T}$ and $\tilde{\mu }$. The color scale represents the peak amplitude of the power spectrum $⟨|\rho (q_{\mathrm{pk}}){|}^2⟩$ for the measured data. Shaded bands indicate the data used for the line graphs in the inset. (Inset) Peak amplitude versus $\tilde{T}$ for (blue circles) $\tilde{\mu }>4.5$, (green diamonds) $\tilde{\mu }\sim 3.4$, and (red squares) $\tilde{\mu }\sim 2.3$. All error bars show the statistical uncertainty as two standard deviations from the mean, and are obtained by bootstrapping.

Figure 3

Dependency of the amplitude of the density ripple power spectrum $⟨|\rho (q_{\mathrm{pk}}){|}^2⟩$ on the relative number of 1D excitations $N_{1\mathrm{D}}/N$. Error bars for the data points represent the statistical uncertainty as two standard deviations from the mean, and are obtained by bootstrapping.