Phase transition dimensionality crossover from two to three dimensions in a trapped ultracold atomic Bose gas

The equilibrium properties of a weakly interacting atomic Bose gas across the Berezinskii-Kosterlitz-Thouless (BKT) and Bose-Einstein condensation (BEC) phase transitions are numerically investigated through a dimensionality crossover from two to three dimensions. The crossover is realized by confining the gas in an experimentally feasible hybridized trap which provides homogeneity along the planar $xy$ directions through a box potential in tandem with a harmonic transverse potential along the transverse $z$ direction. The dimensionality is modified by varying the frequency of the harmonic trap from tight to loose transverse trapping. Our findings, based on a stochastic (projected) Gross-Pitaevskii equation, showcase a continuous shift in the character of the phase transition from BKT to BEC, and a monotonic increase of the identified critical temperature as a function of dimensionality, with the strongest variation exhibited for small chemical potential values up to approximately twice the transverse confining potential.

Figure 1

Examples of isosurface rendering (green) of c-field density ${\left|\mathrm{\Psi }\right|}^2$ during a quench from zero-field conditions to equilibrium for $\mathrm{\Lambda }=1,20,50$ as labeled by (a), (b), and (c), respectively. The temperature of the gas in all cases is 20 nK, which is well below the transition temperature where the system is superfluid. In red we plot an isosurface of high velocity regions, indicating vortex structures. In the $\mathrm{\Lambda }=50$ case, integrated line density profiles are overlayed for each axis. The image at the bottom is the column density, i.e., the density integrated along $z$, for the case of $\mathrm{\Lambda }=50$. The images on the left and right are the column densities in the planar directions.

Figure 2

Results for $\mathrm{\Lambda }=1$ (2D limit). (a) Equilibrium condensate fraction $N_0/N$ as a function of rescaled temperature. The inset plots the quantity $\zeta =(N_{\mathrm{QC}}-N_0)/N_{\mathrm{QC}}$, which highlights the difference between quasicondensate and condensate density as a function of rescaled temperature. (b) Equilibrium ratio of lowest modes $N_1$ and $N_0$ as a function of rescaled temperature. (c) Equilibrium Binder cumulant $C_B$ as calculated from the Penrose-Onsager condensate as a function of rescaled temperature. In each plot, temperature has been rescaled to the BKT critical temperature (8) for an infinite 2D system and marked as a vertical dashed line.

Figure 3

Results for $\mathrm{\Lambda }=50$ (3D limit). (a) Condensate fraction $N_0/N$, alongside the ideal gas law result (dashed magenta line). The inset plots the quantity $\zeta =(N_{\mathrm{QC}}-N_0)/N_{\mathrm{QC}}$, which highlights the difference between quasicondensate and condensate density. (b) Ratio of lowest modes $N_1$ and $N_0$. (c) Binder cumulant $C_B$. In each plot, temperature has been rescaled to the ideal gas BEC transition temperature defined in Eq. (10) and marked as a vertical dashed line.

Figure 4

Transverse profile of the total density of the gas for varying dimensionality $\mathrm{\Lambda }$ at a fixed temperature $T=50$ nK. The coordinate $z$ is normalized to the harmonic oscillator length $l_z$. Note that the curves for $\mathrm{\Lambda }=1,2,3$ are not visible, as they coincide with the one for $\mathrm{\Lambda }=5$. Insets: Zoomed-in profiles for extremes of $\mathrm{\Lambda }=1$ (left) and $\mathrm{\Lambda }=50$ (right) where dashed lines correspond to expected Gaussian and Thomas-Fermi ground-state profiles respectively.

Figure 5

Dependence of equilibrium parameters characterizing the degree of phase coherence of the system as a function of absolute temperature, plotted for different values of the dimensionality parameter $\mathrm{\Lambda }$. (a) Equilibrium condensate fraction $N_0/N$. (b) Ratio of lowest modes $N_1$ and $N_0$. (c) Binder cumulant $C_B$. (d) Order parameter $m$ normalized to the zero-temperature result $m_0$. The dashed horizontal line in each subplot corresponds to a cutoff value used to identify the transition for the respective equilibrium observable (with details of such choice discussed in Appendix pp3). Each color corresponds to a different value of dimensionality $\mathrm{\Lambda }$ as indicated in the legend. Solid vertical colored lines in each subplot correspond to the first numerical point deemed to have crossed the phase transition for the respective quantity towards the incoherent regime, thus marking the identification of the critical temperature for each value of $\mathrm{\Lambda }$ based on that physical quantity. Vertical colored bands (where present) are identical throughout (a)–(d) and indicate the full range of numerically identified critical temperature values across the different quantities plotted in (a)–(d). The midpoint of such band is subsequently chosen as our numerically identified critical temperature for each value of $\mathrm{\Lambda }$.

Figure 6

Numerically extracted phase transition temperature $T_c$ (main plots), and central densities (insets) as a function of dimensionality parameter $\mathrm{\Lambda }$, for two different protocols for our choice of chemical potential $\mu$ based on Eq. (4). Specifically, they correspond to (see text): (a) a constant ${\mu }_{2\mathrm{D}}=\left(4.64\mu {\mathrm{m}}^{-2}\right){\hslash }^2/m$ value, and (b) a value linearly interpolated between such ${\mu }_{2\mathrm{D}}$ value, for $\mathrm{\Lambda }=1$, and $(3/2){\mu }_{2\mathrm{D}}$, for our most 3D case with $\mathrm{\Lambda }=50$. In both cases, black filled points correspond to the mean extracted transition temperature as determined by the equilibrium statistics (Fig. 5), with dashed black lines as a guide to the eye. Error bars and shaded regions highlight the associated uncertainty, arising from a combination of the width of the bands in Fig. 5 and a systematic uncertainty of $\pm 5\mathrm{nK}$ due to our limited resolution in probing distinct temperatures. Hollow blue diamonds mark the analytical $T_{\mathrm{BKT}}$ transition temperature at $\mathrm{\Lambda }=1$, from Eq. (8). Red hollow points for $\mu /\hslash {\omega }_z>3$, with dotted red lines as a guide to the eye, indicate the analytical 3D ideal gas temperature $T_{\mathrm{BEC}}$ in our chosen geometry, using the same total atom number $N$ in Eq. (10) as in the corresponding SPGPE simulation for the same $\mathrm{\Lambda }$. Green points in (a) correspond to further simulations with two different values of $\mu$ at small $\mathrm{\Lambda }$ aimed at demonstrating the limited sensitivity of the rapid initial growth of $T_c$ with $\mathrm{\Lambda }$ on density. Insets: Total density at the center of the trap; for each $\mathrm{\Lambda }$, the point represent an average of the central density in a few SPGPE simulations for $T\sim T_c$, and the colored bands indicate the corresponding numerical uncertainty.

Figure 7

Total ($n$), condensate ($n_0$), and noncondensate ($n_{\text{nc}}=n-n_0$) density profiles along the transverse direction plotted for a quasi-2D ($\mathrm{\Lambda }=10$) and 3D ($\mathrm{\Lambda }=50$) trapping geometry at temperature $T=200$ nK, for which $T/T_c=0.83$ and $T/T_c=0.75$, respectively.