Growth dynamics of a Bose-Einstein condensate in a dimple trap without cooling

We study the formation of a Bose-Einstein condensate in a cigar-shaped three-dimensional harmonic trap, induced by the controlled addition of an attractive “dimple” potential along the weak axis. In this manner we are able to induce condensation without cooling due to a localized increase in the phase-space density. We perform a quantitative analysis of the thermodynamic transformation in both the sudden and adiabatic regimes for a range of dimple widths and depths. We find good agreement with equilibrium calculations based on self-consistent semiclassical Hartree-Fock theory describing the condensate and thermal cloud. We observe that there is an optimal dimple depth that results in a maximum in the condensate fraction. We also study the nonequilibrium dynamics of condensate formation in the sudden turn-on regime, finding good agreement for the observed time dependence of the condensate fraction with calculations based on quantum kinetic theory.

Figure 1

Schematic of the experiment. We create a dimple potential by tightly focusing a laser sheet at the center of the cigar-shaped harmonic magnetic potential. The total external potential $V_{\mathrm{ext}}(z)$ along the axial dimension then consists of a wide harmonic potential plus a narrow Gaussian dimple potential, while the potential in the other two spatial dimensions is unaffected.

Figure 2

Comparison of theory and experiment for (a) the final temperature and (b) the condensate fraction at final equilibrium after the turn-on of the wide dimple potential. Quasistatic turn-on is indicated by solid lines for theory and by diamonds for experimental data. Sudden turn-on is indicated by dashed lines for theory and by squares for experimental data. All experimental data points are four-measurement averages with the error bars indicating the standard deviation of the mean.

Figure 3

Comparison of theory and experiment for (a) the final temperature and (b) the condensate fraction at final equilibrium after the turn-on of the narrow dimple potential. Quasistatic turn-on is indicated by solid lines for theory and by diamonds for experimental data. Sudden turn-on is indicated by dashed lines for theory and by squares for experimental data. The dotted lines indicate quasistatic turn-on from the same initial conditions used for sudden turn-on. All experimental data points are single measurements, with the exception of a four-point measurement at depth $A/k_{\mathrm{B}}=1610$ nK to measure the standard deviation of the mean.

Figure 4

Schematic of the effect of dimple turn-on for small depths $A$. Before the dimple is turned on (solid lines), as the system temperature is above $T_{\mathrm{c}}$ the chemical potential $\mu$ lies below the translational ground-state energy ${\epsilon }_0$ of the harmonic trap. When the dimple is turned on (dashed lines), the new ground-state energy ${\epsilon }_0^{\prime }$ lies below the initial chemical potential, inducing condensation. Due to interactions, the energy ${\epsilon }_0^{\prime }$ increases as the condensate grows until equilibrium is reached (${\epsilon }_0^{\prime }\approx \mu$).

Figure 5

Energy density $n(\epsilon ,t)=g(\epsilon ,t)f(\epsilon ,t)$ before dimple turn-on at $t<0$ (solid line), immediately after dimple turn-on at $t=0$ (dashed line), and at final equilibrium after $t=1000$ ms (dotted line). Strictly speaking, the value plotted at $\epsilon =-A$ is the condensate occupation, not the energy density. Immediately after dimple turn-on to depth $A$, only a small fraction of atoms have energies below the bottom of the harmonic potential ($\epsilon <0$). Over the 1000-ms equilibration time, this fraction steadily increases, eventually resulting in the formation of a condensate.

Figure 6

Condensate growth curves for theory and experiment following the sudden turn-on of the wide dimple potential. The condensate fraction is plotted vs time for dimple depths of $A/k_{\mathrm{B}}=70$ nK (solid line: theory; diamonds: experiment) and $A/k_{\mathrm{B}}=140$ nK (dashed line: theory; squares: experiment). All experimental data points are four-measurement averages with the error bars indicating the standard deviation of the mean.

Figure 7

Condensate growth curves for theory and experiment following the sudden turn-on of the narrow dimple potential. The condensate fraction is plotted vs time for dimple depths of $A/k_{\mathrm{B}}=660$ nK (solid line: theory; diamonds: experiment). All experimental data points are four-measurement averages with the error bars indicating the standard deviation of the mean.

Figure 8

The effects of three-body loss on the predicted condensate fraction. (a) Semiclassical Hartree-Fock predictions of the equilibrium condensate fraction vs dimple depth in a narrow dimple for adiabatic turn on, including three-body loss (solid line), adiabatic turn-on, with no three-body loss (dotted line), sudden turn-on, including three-body loss (dashed line), and sudden turn-on, with no three body loss (dot-dash line). (b) Quantum kinetic theory predictions of condensate growth following the sudden turn-on of the narrow dimple of depth $A/k_{\mathrm{B}}=660\hspace{0.5em}\text{nK}$, including three-body loss (solid line) and with no three-body loss (dashed line).