Abstract
When atoms in a gas are cooled to extremely low temperatures, they will—under the appropriate conditions—condense into a single quantummechanical state known as a Bose–Einstein condensate. In such systems, quantummechanical 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.
Main
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 meanfield theory^{1}, in which the strength of the interactions depends on the atom density and on one additional parameter called the swave 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 zeropoint 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, N_{0}, is less than a critical value given by^{2}
where the dimensionless constant k is called the stability coefficient. The precise value of k depends on the ratio of the magnetic trap frequencies^{3}. a_{ho} is the harmonic oscillator length, which sets the size of the condensate in the idealgas (a = 0) limit.
Under most circumstances, a is insensitive to external fields. But in the vicinity of a socalled Feshbach resonance, a can be tuned over a very large range by adjusting the externally applied magnetic field^{4,5}. This has been demonstrated in recent years with cold ^{85}Rb and Cs atoms^{6,7,8}, and with BECs of Na and ^{85}Rb (refs 9, 10). For ^{85}Rb 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 ^{85}Rb (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 a_{cr} = ka_{ho}/N_{0}. 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 interest^{12,13,14,15,16,17} was generated by BEC experiments in ^{7}Li (ref. 18), for which the scattering length is also negative and collapse events are also observed^{19,20}. The ^{7}Li 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 N_{cr}, and the collapse is driven by a stochastic process. In addition, studies of collapse dynamics in ^{7}Li 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 N_{0} is much larger than N_{cr}, allows us to explore the dynamics and compare it with theory in far more detail.
Experimental techniques
The procedure for producing stable ^{85}Rb condensates has been described in detail elsewhere^{10}. A standard double magnetooptical trap (MOT) system^{21} was used to collect a cold sample of ^{85}Rb atoms in a lowpressure chamber. Once sufficient atoms had accumulated in the lowpressure MOT, the atoms were loaded into a cylindrically symmetric cigarshaped magnetic trap with frequencies ν_{radial} = 17.5 Hz and ν_{axial} = 6.8 Hz. Radiofrequency 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 welldefined initial condition, with the BEC taking on the size and shape of the harmonic oscillator ground state.
We could then adjust the meanfield 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 < a_{cr} 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 fullwidth at halfmaximum). 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 preexpansion 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 a_{collapse} < a_{cr}, 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 densitydependent inelastic collisions^{22} 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 T_{rad}/4 ≈ 14 ms, where T_{rad} 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 τ_{evolve} for a_{collapse} = 30a_{0} and a_{init} = +7a_{0} is shown in Fig. 1b. The number remained constant for some time after the jump in a until atom loss suddenly began at t_{collapse}. 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 meanfield expansion. We observed that the postexpansion condensate widths changed very little with time τ_{evolve} before t_{collapse}. 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 a_{collapse} 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 × 10^{13} cm^{3}. Using the decay constants from ref. 22, this density gives an atom loss rate, τ_{decay}, 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 a_{quench} = 0 after a time τ_{evolve} at a_{collapse}. 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 a_{quench} = 0 or to a_{quench} = 250a_{0}.
We have measured loss curves like that in Fig. 1b for many different values of a_{collapse}. The collapse time versus a_{collapse} for N_{0} = 6,000 presented in Fig. 2 shows a strong dependence on a_{collapse}. Reducing the initial density by a factor of ∼4 (with a corresponding increase in volume) by setting a_{init} = +89a_{0} for one value of a_{collapse} (15a_{0}), increased t_{collapse} by a factor of about three.
The atom loss time constant τ_{decay} depended only weakly on a_{collapse} and N_{0}. For the range of a_{collapse} shown in Fig. 2, τ_{decay} did not depend on a_{collapse} or N_{0} outside of the experimental noise (∼20%). On average, τ_{decay} was 2.8(1) ms. For the very negative value of a_{collapse} ≈ 250a_{0}, however, τ_{decay} did decrease to 1.8 ms for N_{0} = 6,000 and to 1.2 ms for N_{0} = 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, spinpolarized 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 a_{collapse} and N_{0} are complex, but as they will provide a stringent test of collapse theories, we present them in detail.
The angular kineticenergy 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 (T_{rad}/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 crosssections 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 a_{collapse}. 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 welldefined and showed trends far larger than the variation. The axial and radial burst energies versus a_{collapse} are shown in Fig. 4a and b for N_{0} = 6,000 and 15,000, respectively. The burstenergy anisotropy shown in Fig. 4c depended on N_{0}, a_{collapse} and a_{init}.
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 N_{burst} grew with τ_{evolve} 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. N_{burst} 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 N_{0} and did not depend on a_{collapse} or N_{0}.
Remnant condensate.
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 ≈ 2ν_{axial} and ν ≈ 2ν_{radial}) 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 a_{collapse}.
The number of atoms in the remnant depended on a_{collapse} and N_{0}, and in general was not limited by the critical number, N_{cr}. The stability condition in equation (1) determined the collapse point, but did not constrain N_{remnant}. A fixed fraction of N_{0} went into the remnant independent of N_{0}, so that smaller condensates often ended up with N_{remnant} < N_{cr}, but larger condensates rarely did. The fraction of atoms that went into the remnant decreased with a_{collapse}, and was ∼40% for a_{collapse} < 10a_{0} and ∼10% for a_{collapse} > 100a_{0}. We do not think that surfacewave excitations^{24} are responsible for stabilizing the remnant because we excite largeamplitude breathing modes. For N_{0} = 6,000 and a_{collapse} < 10a_{0}, more atoms were lost than the number required to lower N_{remnant} to below N_{cr}.
As N_{burst} was independent of a_{collapse} but N_{remnant} decreased with a_{collapse}, the number of missing atoms increased with a_{collapse}. The number of missing atoms also increased with N_{0}, but the percentage of missing atoms was equal for N_{0} = 6,000 and N_{0} = 15,000 with a_{collapse} fixed and was ∼40% for a_{collapse} < 10a_{0} and ∼70% for a_{collapse} ≥ 100a_{0}. 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.
Jet formation.
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 a_{quench} = 0) during the period of number loss. Collapse processes that were allowed to evolve to completion (until N ≈ N_{remnant}) were not observed to emit jets. Examples are shown in Fig. 5 for different τ_{evolve} 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 a_{quench} = 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)mv^{2} = ℏ^{2}/(4mσ^{2}), 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 τ_{evolve} 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 papers^{12,13,14,15,16,17} have considered the problem of collapse of a BEC when the number of atoms exceeds the critical number. These treatments all use a meanfield approach, and describe the condensate dynamics using the Gross–Pitaevskii (GP) equation. In most cases^{12,13,15,16}, the loss mechanism is threebody recombination, but Duine and Stoof^{17} 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 threedimensional anisotropic timedependent 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 Shlyapnikov^{13} numerically integrate the GP equation for the case of an isotropic trap and large values of the threebody recombination coefficient K_{3}. In this regime, the condensate smoothly contracts in a single, collective collapse. Saito and Ueda^{15,16} perform a similar numerical solution for the isotropic case but with smaller values of K_{3}, and observe localized spikes to form in the wavefunction during collapse. Duine and Stoof^{17} 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.
Theoretical challenges.
Collapsing ^{85}Rb condensates are simple systems with dramatic behaviour. This behaviour might provide a rigorous test of meanfield theory when it is applied to our experimental conditions. Some of our particularly puzzling results are as follows. (1) The decay constant τ_{decay} is independent of both N_{0} and a_{collapse} for a_{collapse} < 100a_{0}, and only weakly depends on these quantities for larger a_{collapse}. (2) The burst energy per atom dramatically increases with initial condensate number. (3) The fraction of burst atoms is constant when a_{collapse} changes. (4) The number of cold remnant BEC atoms surviving the collapse varies between much less and much more than N_{cr}, depending on N_{0} and a_{collapse}, but the fractions of remnant atoms, burst atoms, and missing atoms are independent of N_{0}.
Outlook
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 highenergy 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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Acknowledgements
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, AROMURI and NIST. S.L.C. acknowledges the support of a Lindemann Fellowship.
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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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DOI: https://doi.org/10.1038/35085500
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