When atoms in a gas are cooled to extremely low temperatures, they will—under the appropriate conditions—condense into a single quantum-mechanical state known as a Bose–Einstein condensate. In such systems, quantum-mechanical behaviour is evident on a macroscopic scale. Here we explore the dynamics of how a Bose–Einstein condensate collapses and subsequently explodes when the balance of forces governing its size and shape is suddenly altered. A condensate's equilibrium size and shape is strongly affected by the interatomic interactions. Our ability to induce a collapse by switching the interactions from repulsive to attractive by tuning an externally applied magnetic field yields detailed information on the violent collapse process. We observe anisotropic atom bursts that explode from the condensate, atoms leaving the condensate in undetected forms, spikes appearing in the condensate wavefunction and oscillating remnant condensates that survive the collapse. All these processes have curious dependences on time, on the strength of the interaction and on the number of condensate atoms. Although the system would seem to be simple and well characterized, our measurements reveal many phenomena that challenge theoretical models.
Although the density of an atomic Bose–Einstein condensate (BEC) is typically five orders of magnitude lower than the density of air, the interatomic interactions greatly affect a wide variety of BEC properties. These include static properties, such as the size, shape and stability of the condensate, and dynamic properties, like the collective excitation spectrum and soliton and vortex behaviour. Because all of these properties are sensitive to the interatomic interactions, they can be dramatically affected by tuning the strength and sign of the interactions.
Most of the physical processes in BECs are well described by mean-field theory1, in which the strength of the interactions depends on the atom density and on one additional parameter called the s-wave scattering length, a. The parameter a is determined by the atomic species. When a > 0, the interactions are repulsive. In contrast, when a < 0, the interactions are attractive and a BEC tends to contract to minimize its overall energy. In a magnetic trap, the contraction competes with the kinetic zero-point energy, which tends to spread out the condensate. For a strong enough attractive interaction, there is not enough kinetic energy to stabilize the BEC, and it is expected to implode. A BEC can avoid implosion only as long as the number of atoms in it, N0, is less than a critical value given by2
where the dimensionless constant k is called the stability coefficient. The precise value of k depends on the ratio of the magnetic trap frequencies3. aho is the harmonic oscillator length, which sets the size of the condensate in the ideal-gas (a = 0) limit.
Under most circumstances, a is insensitive to external fields. But in the vicinity of a so-called Feshbach resonance, a can be tuned over a very large range by adjusting the externally applied magnetic field4,5. This has been demonstrated in recent years with cold 85Rb and Cs atoms6,7,8, and with BECs of Na and 85Rb (refs 9, 10). For 85Rb atoms, a is usually negative, but a Feshbach resonance at ∼155 G allows us to tune a by orders of magnitude and even change its sign. This gives us the ability to create stable BECs of 85Rb (ref. 10) and to adjust the interatomic interactions. We recently used this flexibility to verify the functional form of equation (1), and to measure the stability coefficient to be k = 0.46(6) (ref. 11).
Here we study the dynamical response (‘the collapse’) of an initially stable BEC to a sudden shift of the scattering length to a value more negative than the critical value acr = -kaho/N0. We have observed many features of the surprisingly complex collapse process, including the energies and energy anisotropies of atoms that burst from the condensate, the timescale for the onset of this burst, the rates for losing atoms, spikes in the wavefunction that form during collapse, and the size of the remnant BEC that survives the collapse. The level of control provided by tuning a has allowed us to see how all of these quantities depend on the magnitude of a, on the initial number and density of condensate atoms, and on the initial spatial size and shape of the BEC before the transition to instability.
A great deal of theoretical interest12,13,14,15,16,17 was generated by BEC experiments in 7Li (ref. 18), for which the scattering length is also negative and collapse events are also observed19,20. The 7Li experiments do not use a Feshbach resonance, so a is fixed. This restricts experimentation to the regime where the initial number of condensate atoms is less than or equal to Ncr, and the collapse is driven by a stochastic process. In addition, studies of collapse dynamics in 7Li are complicated by a large thermal component. Our ability to tune the scattering length, and to explore the regime where the initial condensate is ‘pure’ (near T = 0) and the number N0 is much larger than Ncr, allows us to explore the dynamics and compare it with theory in far more detail.
The procedure for producing stable 85Rb condensates has been described in detail elsewhere10. A standard double magneto-optical trap (MOT) system21 was used to collect a cold sample of 85Rb atoms in a low-pressure chamber. Once sufficient atoms had accumulated in the low-pressure MOT, the atoms were loaded into a cylindrically symmetric cigar-shaped magnetic trap with frequencies νradial = 17.5 Hz and νaxial = 6.8 Hz. Radio-frequency evaporation was then used to cool the sample to ∼3 nK to form pure condensates containing >90% of the sample atoms. The final stages of evaporation were performed at 162 G, where the scattering length is positive and stable condensates of up to 15,000 atoms could be formed. After evaporative cooling, the magnetic field was increased adiabatically to 166 G (except where noted), where a = 0. This provided a well-defined initial condition, with the BEC taking on the size and shape of the harmonic oscillator ground state.
We could then adjust the mean-field interactions within the BEC to a variety of values on timescales as short as 0.1 ms. The obvious manipulation was to jump to some value of a < acr to trigger a collapse, but the tunability of a was also a great help in imaging the sample. Usually the condensate size was below the resolution limit of our imaging system (7 µm full-width at half-maximum). However, we could increase the scattering length to large positive values and use the repulsive interatomic interactions to expand the BEC before imaging, thus obtaining information on the pre-expansion condensate shape and number. A typical a(t) sequence is shown in Fig. 1a. We have used a variety of such sequences to explore many aspects of the collapse, and enhance the visibility of particular components of the sample.
Condensate contraction and atom loss
When the scattering length is changed rapidly to a value acollapse < acr, a condensate's kinetic energy no longer provides a sufficient barrier against collapse. As described in ref. 13, during collapse one might expect a BEC to contract until losses from density-dependent inelastic collisions22 would effectively stop the contraction. This contraction would take place roughly on the timescale of a trap oscillation, and the density would sharply increase after Trad/4 ≈ 14 ms, where Trad is the radial trap period. How does this picture compare to what we have seen?
A plot of the number of atoms in the condensate, N, versus τevolve for acollapse = -30a0 and ainit = +7a0 is shown in Fig. 1b. The number remained constant for some time after the jump in a until atom loss suddenly began at tcollapse. After the jump the condensate was smaller than our resolution limit, so we could not observe the contraction directly, but we could infer the extent of the contraction by observing the degree of mean-field expansion. We observed that the post-expansion condensate widths changed very little with time τevolve before tcollapse. From this we infer that the bulk BEC did not contract dramatically before loss began. The equations in ref. 23 (in the gaussian approximation) predict fairly well the observed expanded shape over a fairly large range of acollapse and t before collapse. So we used these equations to estimate the density before collapse, and find that the predicted contraction only corresponds to a 50% increase in the average density to 2.5 × 1013 cm-3. Using the decay constants from ref. 22, this density gives an atom loss rate, τdecay, that is far smaller than what we observe and does not have the observed sudden onset.
For the data in Fig. 1b and most other data presented below, a was changed (in 0.1 ms) to aquench = 0 after a time τevolve at acollapse. We believe that the loss stopped immediately after this jump in a. This interpretation is based on the observation that the quantitative details of curves such as that shown in Fig. 1b did not depend on whether the collapse was terminated by a jump to aquench = 0 or to aquench = 250a0.
We have measured loss curves like that in Fig. 1b for many different values of acollapse. The collapse time versus acollapse for N0 = 6,000 presented in Fig. 2 shows a strong dependence on acollapse. Reducing the initial density by a factor of ∼4 (with a corresponding increase in volume) by setting ainit = +89a0 for one value of acollapse (-15a0), increased tcollapse by a factor of about three.
The atom loss time constant τdecay depended only weakly on acollapse and N0. For the range of acollapse shown in Fig. 2, τdecay did not depend on acollapse or N0 outside of the experimental noise (∼20%). On average, τdecay was 2.8(1) ms. For the very negative value of acollapse ≈ -250a0, however, τdecay did decrease to 1.8 ms for N0 = 6,000 and to 1.2 ms for N0 = 15,000.
Bursts of atoms.
Atoms leave the BEC during the collapse (Fig. 1b). There are at least two components to the expelled atoms. One component (the ‘missing atoms’) is not detected. The other component emerges as a burst of detectable, spin-polarized atoms with energies much greater than the initial energy of the condensate but much less than the magnetic trap depth. The dependences of burst energy on acollapse and N0 are complex, but as they will provide a stringent test of collapse theories, we present them in detail.
The angular kinetic-energy distribution with which the burst atoms are expelled from the condensate can most accurately be measured by observing their harmonic oscillations in the trap (Fig. 3a). For example, half of a radial period after the expulsion (Trad/2), all atoms return to their initial radial positions. Well before or well after this ‘radial focus’, the burst cloud is too dilute to be observed. Fortunately, at the radial focus, oscillations along the axial trap axis are near their outer turning points, and the axial energy can be found from the length of the ‘stripe’ of atoms along the axial axis. The radial energy can be found with the same procedure for an axial focus. The sharpness of the focus also provides information on the time extent of the burst.
Figure 3b shows an image of a radial focus. The size scales for the burst focus and the remnant were well separated because the latter was not expanded before imaging. Figure 3c shows cross-sections of the burst focus, and fits to the burst and to the thermal cloud. The burst energy distributions were well fitted by gaussians characterized by a temperature that was usually different for the two trap directions. The burst energy fluctuated from shot to shot by up to a factor of 2 for a given acollapse. This variation is far larger than the measurement uncertainty or the variation in initial number (both ∼10%), and its source is unknown. (We also discuss observed structural variability when we present the jet measurements below.)
Although the burst energies varied from shot to shot, the average value was well-defined and showed trends far larger than the variation. The axial and radial burst energies versus acollapse are shown in Fig. 4a and b for N0 = 6,000 and 15,000, respectively. The burst-energy anisotropy shown in Fig. 4c depended on N0, acollapse and ainit.
When we interrupted the collapse with a jump back to a = 0 as discussed above, we also interrupted the growth of the burst. The ‘interrupted’ burst atoms still refocused after sitting at a = 0 for the requisite half trap period. The energy of the atoms in the interrupted bursts appeared to be the same, but the number of atoms was smaller. By changing the time at which the collapse was interrupted, we could measure the time dependence of the creation of burst atoms. For the conditions of Fig. 1b, the number of burst atoms Nburst grew with τevolve with a time constant of 1.2 ms, starting at 3.5 ms and reaching an asymptotic final number of ∼2,500 for all times ≥7 ms. Nburst varied randomly by ∼20% for the data in Fig. 4, but on average the fraction of atoms going into the burst was about 20% of N0 and did not depend on acollapse or N0.
After a collapse, a ‘remnant’ condensate containing a fraction of the atoms survived with nearly constant number for more than 1 second, and oscillated in a highly excited collective state with the two lowest modes (ν ≈ 2νaxial and ν ≈ 2νradial) being predominantly excited. (The measured frequencies were ν = 13.6(6) Hz and ν = 33.4(3) Hz.) To find the oscillation frequencies, the widths of the condensate were measured as a function of time spent at acollapse.
The number of atoms in the remnant depended on acollapse and N0, and in general was not limited by the critical number, Ncr. The stability condition in equation (1) determined the collapse point, but did not constrain Nremnant. A fixed fraction of N0 went into the remnant independent of N0, so that smaller condensates often ended up with Nremnant < Ncr, but larger condensates rarely did. The fraction of atoms that went into the remnant decreased with |acollapse|, and was ∼40% for |acollapse| < 10a0 and ∼10% for |acollapse| > 100a0. We do not think that surface-wave excitations24 are responsible for stabilizing the remnant because we excite large-amplitude breathing modes. For N0 = 6,000 and |acollapse| < 10a0, more atoms were lost than the number required to lower Nremnant to below Ncr.
As Nburst was independent of acollapse but Nremnant decreased with |acollapse|, the number of missing atoms increased with |acollapse|. The number of missing atoms also increased with N0, but the percentage of missing atoms was equal for N0 = 6,000 and N0 = 15,000 with acollapse fixed and was ∼40% for |acollapse| < 10a0 and ∼70% for |acollapse| ≥ 100a0. The missing atoms were presumably either expelled from the condensate at such high energies that we could not detect them (>20 µK), or they were transferred to untrapped atomic states or undetected molecules.
Under very specific experimental conditions, we observed streams of atoms with highly anisotropic velocities emerging from the collapsing condensate. These ‘jets’ are distinguished from the ‘burst’ in that the jets have much lower kinetic energy (on the order of a few nanokelvin), in that their velocity is nearly purely radial, and in that they appear only when the collapse is interrupted (that is, by jumping to aquench = 0) during the period of number loss. Collapse processes that were allowed to evolve to completion (until N ≈ Nremnant) were not observed to emit jets. Examples are shown in Fig. 5 for different τevolve for the conditions of Fig. 1b. The size and shape of the jets varied from image to image even when all conditions were unchanged, and as many as three jets were sometimes observed to be emitted from the collapse of a single condensate. Also, the jets were not always symmetric about the condensate axis.
We believe that these jets are manifestations of local ‘spikes’ in the condensate density that form during the collapse and expand when the balance of forces is changed by quenching the collapse. We can estimate the size of the spikes using the uncertainty principle. After a jump to aquench = 0, the kinetic energy per atom in the resulting jet is equal to the confinement energy that the spike had before quenching the collapse, that is, (1/2)mv2 = ℏ2/(4mσ2), where σ is the width of the spike in the wavefunction, m is the mass, and v is the expansion velocity. The anisotropy of the jets indicates that the spikes from which they originated were also highly anisotropic, being narrower in the radial direction. From the widths and the number of atoms in the jets, we can estimate the density in the spikes. Plots of the number of jet atoms and the inferred density in the spikes versus τevolve are presented in Fig. 6. The jets exhibited variability in energy and number that was larger than the ∼10% measurement noise.
Overview of the current theoretical models
Several theoretical papers12,13,14,15,16,17 have considered the problem of collapse of a BEC when the number of atoms exceeds the critical number. These treatments all use a mean-field approach, and describe the condensate dynamics using the Gross–Pitaevskii (GP) equation. In most cases12,13,15,16, the loss mechanism is three-body recombination, but Duine and Stoof17 propose that the loss arises from a new elastic scattering process. In both cases the loss is density dependent, and so the loss rate is quite sensitive to the dynamics of the shape of the condensate. Because a full three-dimensional anisotropic time-dependent solution to the GP equation is very difficult, these calculations have used various approximations to calculate the time evolution of the condensate shape. Kagan, Muryshev and Shlyapnikov13 numerically integrate the GP equation for the case of an isotropic trap and large values of the three-body recombination coefficient K3. In this regime, the condensate smoothly contracts in a single, collective collapse. Saito and Ueda15,16 perform a similar numerical solution for the isotropic case but with smaller values of K3, and observe localized spikes to form in the wavefunction during collapse. Duine and Stoof17 model the dynamics for the anisotropic case, but use a gaussian approximation rather than an exact numerical solution.
These calculations have all been performed over a certain range of parameters, but none have been done for the specific range of parameters that correspond to our experimental situations. None of the predictions in these papers match our measurements except for the general feature that atoms are lost from the condensate. Also, we see several phenomena that are not discussed in these papers. Whether this lack of agreement is due to the fact that these calculations do not scale to our experimental situation or do not contain the proper physics remains to be seen.
Collapsing 85Rb condensates are simple systems with dramatic behaviour. This behaviour might provide a rigorous test of mean-field theory when it is applied to our experimental conditions. Some of our particularly puzzling results are as follows. (1) The decay constant τdecay is independent of both N0 and |acollapse| for |acollapse| < 100a0, and only weakly depends on these quantities for larger |acollapse|. (2) The burst energy per atom dramatically increases with initial condensate number. (3) The fraction of burst atoms is constant when acollapse changes. (4) The number of cold remnant BEC atoms surviving the collapse varies between much less and much more than Ncr, depending on N0 and acollapse, but the fractions of remnant atoms, burst atoms, and missing atoms are independent of N0.
From the experimental point of view, there are at least two questions to be answered. First, is the burst coherent? It may be possible to answer this by generating a sequence of ‘half bursts’ and see if they interfere. Second, where do the missing atoms go? If molecules and/or relatively high-energy atoms are being created, can we detect them? It is clear that adjustable interactions open up new avenues for BEC studies.
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We thank S. Thompson for laboratory assistance, and S. Dürr, G. Shlyapnikov, H. Stoof, M. Holland, M. Ueda and R. Duine for discussions. This work was supported by the ONR, NSF, ARO-MURI and NIST. S.L.C. acknowledges the support of a Lindemann Fellowship.
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Cite this article
Donley, E., Claussen, N., Cornish, S. et al. Dynamics of collapsing and exploding Bose–Einstein condensates. Nature 412, 295–299 (2001). https://doi.org/10.1038/35085500
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