Macroscopic quantum tunneling escape of Bose-Einstein condensates

Recent experiments on macroscopic quantum tunneling reveal a nonexponential decay of the number of atoms trapped in a quasibound state behind a potential barrier. Through both experiment and theory, we demonstrate this nonexponential decay results from interactions between atoms. Quantum tunneling of tens of thousands of ${}^{87}\mathrm{Rb}$ atoms in a Bose-Einstein condensate is modeled by a modified Jeffreys-Wentzel-Kramers-Brillouin model, taking into account the effective time-dependent barrier induced by the mean field. Three-dimensional Gross-Pitaevskii simulations corroborate a mean-field result when compared with experiments. However, with one-dimensional modeling using time-evolving block decimation, we present an effective renormalized mean-field theory that suggests many-body dynamics for which a bare mean-field theory may not apply.

Figure 1

Macroscopic quantum tunneling experiment. (a) Schematic of experimental time sequence to obtain MQT in a BEC. (b) Experimental 3D potential with a barrier height of 190 nK (peak height) in the weak configuration. The distance between saddle points to the potential minimum is $x_0=18\left(1\right)\mu \mathrm{m}$. Sketched is a BEC trapped in the local minimum of the potential (purple ellipse) escaping via the weakest part of the potential (purple arrows). (c) Experimental data, in number of trapped atoms $N$, exhibits a nonexponential decay, in contrast to the exponential background decay due to atomic losses (pink nearly horizontal dashed line). The vertical dashed lines divide the experimental dynamics into the three dominant regions of classical spilling, MQT, and background decay.

Figure 2

Considerations beyond single-particle tunneling. (a) Interatomic interaction reforms a square barrier, creating an effective potential (dashed lines). (b) Statistical properties of particles in Tonks-Girardeau gas, fermion (upper red curve) vs boson (lower blue curve), produce different short-time deviations in particle number from single-particle exponential decay. (c) Trajectories of atoms in the trapping potential can become chaotic for many geometries including ours. Despite these complications, effective 1D models can still be useful, as we demonstrate in this article.

Figure 3

Experimental and numerical 3D Gross-Pitaevskii chemical potential. Chemical potential $\mu$ as a function of the total number of atoms $N$. (a) Weak configuration barrier heights: $V_0=460$(30) nK (red circles), 260(18) nK (blue diamonds), and 170(11) nk (green squares). (b) Tight configuration: $V_0=330$(35) nK (red circles), 290(30) nK (blue diamonds), and 240(25) nK (green squares). Mean-field simulations predict a chemical potential (black line) in agreement with experimental data, using barrier heights (a) 350 nK and (b) 300 nK.

Figure 4

Numerical 3D Gross-Pitaevskii and experimental data. Semilog plot for decay in number of trapped atoms for (a) weak and (b) tight trapping configuration. Data from the classical spilling regime around the first 40 ms (20 ms) in weak (tight) configuration is also included. Gray lines are 3D GPE simulations. Beyond (a) 0.8 s and (b) 0.6 s, the dynamics become loss dominated. (c) 3D simulations reproduce gross features of decay rate as a function of chemical potential, where gray data points are classical spilling transients, and dashed black lines are fitting results. All data points are from the experiment.

Figure 5

Case study: theoretical model fit to macroscopic quantum tunneling data. Experimental number of trapped atoms as a function of time (red circles, mean values with $1\sigma$ error bars) with theoretical fit (solid blue curve), exponential fit through tunneling dominated regime (dash dot black line), and experimental background loss (dashed pink nearly horizontal line). We divide the decay curve into three subregions (indicated by dashed vertical lines): initial transient classical spilling, (A) mean-field assisted quantum tunneling region with nonexponential decay, and (B) background loss dominated region. The green (dark gray) envelope indicates uncertainty in fitting parameters from modified JWKB. The yellow (light gray) envelope indicates combined uncertainty due to uncertainty in experimental parameters, uncertainty in data, and uncertainty in fit parameters.

Figure 6

JWKB potential schematic. Tunneling in the full 3D experimental potential can be modeled with a much simpler 1D approach. The minimal feature set required to fit the data is (1) a bare 1D triangle potential of height $V_{\mathrm{s}}$ and width $2w$ (gold, solid); (2) inclusion of the mean field to create an effective potential of height $V_{\mathrm{s}}(1+a)$ with $a\propto N\left(t\right)$ (teal, dotted); and (3) use of the Thomas-Fermi approximation for the chemical potential $\mu \propto N^{1/3}$ (red dashed line). Such a choice of model provides self-consistency between the replacement of $E$ with $\mu$ in the usual JWKB treatment, together with the bare $V\left(x\right)$ with the effective potential $V_{\mathrm{eff}}\left(x\right)=V\left(x\right)+g{\left|\psi \right|}^2$.

Figure 7

Fitting results in quantum tunneling regime. Optimized values of (a) effective saddle height (teal squares) at start of tunneling dynamics in comparison to the experimentally measured bare saddle height (red triangles) vs the bare peak height, showing how mean-field effects significantly correct the tunneling dynamics. (b) Effective potential width $w$ for the weak configuration. Both the effective width and height increase as a function of bare experimental barrier height $V$.

Figure 8

Experimental data and theoretical curves comparison: decay rate vs chemical potential. (a) Experimental data for the weak trapping configuration (colored points), 3D GPE simulations (gray lines), and fit results to an exponential plus a constant, Eq. (2) (dashed black lines). (b) Modified JWKB curves modeling this data (solid colored lines) and fit curves to our model, again based on Eq. (2) (dashed black lines), showing that the basic exponential dependence on chemical potential is captured by a much simpler 1D modified JWKB approach. Numbers in (a) and (b) represent barrier height $V_{\mathrm{s}}$.

Figure 9

Theoretical fits of trapped particle number for weak configurations. All fits demonstrate the same trends as the case study in Fig. 5. Gray dashed lines represent background loss for each barrier height. Colors indicate barrier heights for weak configuration, in nanokelvin.

Figure 10

Kinetic energy correction to the chemical potential. In the (a) weak and (b) tight configurations, as the number of atoms increases, the kinetic energy contribution decreases, rendering the Thomas-Fermi approximation more accurate. Tight configuration requires kinetic energy a correction due to fewer particle numbers. Colored dots are from 3D GPE calculation while lines are a guide to the eye.

Figure 11

Experimental data and theoretical fits in tight configurations. Experimental data for MQT (theoretical fits, blue lines) for the tight configuration shows either (a) for $V=290$ nK, large experimental error and small fitting error, or (b) for $V=330$ nK, few data points with large fitting error. Red points are the mean value of the number of atoms in the trap from experimental data, with $1\sigma$ error bars, and dashed pink line is background loss. The green regions indicate fitting error, and yellow regions combined uncertainty in model fit, experimental parameters, and data points.

Figure 12

Effective mean field, many body, and depletion. (a) Normalized trapped atoms for many-body $(NU/J=1.00$, dashed curve) is bounded from above and below by effective values $g=0.72$ (green, upper dark gray curve) and $g=0.77$ (dotted curve), respectively, while direct mean-field comparison $g=1.00$ (teal, the lowest dark gray curve) underpredicts. (b) Semilog two largest eigenvalues $\left({\lambda }_1,{\lambda }_2\right)$ and sum of remaining eigenvalues ${\lambda }_{\ge 3}$ of the single-particle density matrix for the total system (solid lines) and trapping well (dashed lines) are plotted versus time. The wave function has large occupation of two modes over the whole system (solid lines), with 10% depletion as noted by nonzero ${\lambda }_{\ge 3}$. Points represent actual data with error bars smaller than marker, and lines are a guide to the eye.