Dynamics of collapsing and exploding Bose–Einstein condensates

Abstract

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.

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 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. a_ho 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 field^4,5. This has been demonstrated in recent years with cold ⁸⁵Rb and Cs atoms^6,7,8, 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 a_cr = -ka_ho/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 interest^{12,13,14,15,16,17} was generated by BEC experiments in ⁷Li (ref. 18), for which the scattering length is also negative and collapse events are also observed^19,20. 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 N_cr, 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 N_cr, allows us to explore the dynamics and compare it with theory in far more detail.

Experimental techniques

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 ν_radial = 17.5 Hz and ν_axial = 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 < 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 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.

Figure 1: An example of a ramp applied to the scattering length a, and a plot of the condensate number N versus time after a jump to a negative scattering length.

a, A typical a(t) sequence. a₀ = 0.529 Å is the Bohr radius. The scattering length is jumped at t = 0 in 0.1 ms from a_init to a_collapse, where the BEC evolves for a time τ_evolve. The field is carefully controlled so that magnetic-field noise translates into fluctuations in a_collapse on the order of ∼0.1a₀ in magnitude. The collapse is then interrupted with a jump to a_quench, and the field is ramped in 5 ms to a large positive scattering length which makes the BEC expand. After 7.5 ms of additional expansion, the trap is turned off in 0.1 ms, and 1.8 ms later the density distribution is probed using destructive absorption imaging with a 40-µs laser pulse (indicated by the vertical bar). The increase in a from a_collapse to a_expand is far too rapid to allow for the BEC to expand adiabatically. On the contrary, the smaller the BEC before expansion, the larger the cloud at the moment of imaging. Thus we can readily infer the relative size of the bulk of the BEC just before the jump to a_quench. The density of the expanded BEC is so low that the rapid transit of the Feshbach resonance pole²⁵ during the trap turn-off and the subsequent time spent at magnetic field B = 0 (a = -400a₀) both have a negligible effect. b, The number of atoms remaining in the BEC versus τ_evolve at a_collapse = -30a₀. We observed a delayed and abrupt onset of loss. The solid line is a fit to an exponential with a best-fit value of t_collapse ≈ 3.7(5) ms for the delay.

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 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 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 = -30a₀ and a_init = +7a₀ 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 mean-field expansion. We observed that the post-expansion 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 × 10¹³ 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 = 250a₀.

We have measured loss curves like that in Fig. 1b for many different values of a_collapse. The collapse time versus a_collapse for N₀ = 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 = +89a₀ for one value of a_collapse (-15a₀), increased t_collapse by a factor of about three.

Figure 2: The collapse time t_collapse versus the scattering length to which the condensate is jumped, a_collapse for 6,000-atom condensates.

The vertical line indicates the critical value of the scattering length, a_cr for N₀ = 6,000. The data were acquired with a_init = a_quench = 0 (to within ∼2a₀; see Fig. 1).

The atom loss time constant τ_decay depended only weakly on a_collapse and N₀. For the range of a_collapse shown in Fig. 2, τ_decay did not depend on a_collapse or N₀ outside of the experimental noise (∼20%). On average, τ_decay was 2.8(1) ms. For the very negative value of a_collapse ≈ -250a₀, however, τ_decay 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 a_collapse 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 (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 3: A burst focus.

a, Conceptual illustration of a radial burst focus. t_burst is the time at which the burst is generated and T_rad is the radial trap period. b, An image of a radial burst focus taken 33.5 ms after a jump from a_init = 0 to -30a₀ for N₀ = 10,000. T_rad/2 = 28.6 ms, which indicates that the burst occurred 4.9(5) ms after the jump. The axial energy distribution for this burst corresponded to an effective temperature of 62 nK. The image is 60 × 310 µm. c, Radially averaged cross-section of b with a gaussian fit to the burst energy distribution. The central 100 µm were excluded from the fit to avoid distortion in the fit due to the condensate remnant (σ = 9 µm) and the thermal cloud (σ = 17 µm). The latter is present in the pre-collapse sample due to the finite temperature, and appears to be unaffected by the collapse. The dashed line indicates the fit to this initial thermal component. We note the offset between the centres of the burst and the remnant. This offset varies from shot to shot by an amount comparable to the offset shown.

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 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 well-defined 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 N₀ = 6,000 and 15,000, respectively. The burst-energy anisotropy shown in Fig. 4c depended on N₀, a_collapse and a_init.

Figure 4: Burst energies and energy anisotropies.

a and b, the axial and radial burst energies versus a_collapse for N₀ ≈ 6,000 and N₀ ≈ 15,000, respectively. On average, ten burst focuses were measured for each trap direction at each value of a_collapse studied. The energies were higher for the larger-N₀ condensates over the full range of a_collapse studied. The vertical and horizontal error bars indicate respectively the standard error of the measurements and the uncertainty in a_collapse arising from the magnetic-field calibration. For several of the points, the uncertainties are smaller than the symbol size and the error bars are not visible. c, The ratio of the radial to the axial energies, E_rad/E_ax, which is a measure of the burst anisotropy. For values of |a_collapse| just past a_cr, the burst was isotropic for both N₀ ≈ 6,000 and N₀ ≈ 15,000. At larger values of |a_collapse|, larger-N₀ condensates gave rise to stronger anisotropies. For N₀ = 6,000, a_init/a₀ was 0 and λ was 1.6, and for N₀ = 15,000, a_init/a₀ was +7 and λ was 2, where λ = σ_z/σ_r is the aspect ratio of the initial condensate. When instead we started at a_init = +100a₀, the BEC was initially more anisotropic (λ = 2.4), but the burst became more isotropic, with E_ax going up by ∼40% and E_rad dropping by ∼60% for N₀ = 15,000 and a_collapse = -100a₀.

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 N₀ and did not depend on a_collapse or N₀.

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 N₀, 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 N₀ went into the remnant independent of N₀, 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| < 10a₀ and ∼10% for |a_collapse| > 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 |a_collapse| < 10a₀, 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 N₀, but the percentage of missing atoms was equal for N₀ = 6,000 and N₀ = 15,000 with a_collapse fixed and was ∼40% for |a_collapse| < 10a₀ and ∼70% for |a_collapse| ≥ 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.

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.

Figure 5: Jet images for a series of τ_evolve values for the conditions of Fig. 1b.

The evolution times were 2, 3, 4, 6, 8 and 10 ms (from a to f). Each image is 150 × 255 µm. The bar indicates the optical depth scale. An expansion to a_expand = +250a₀ was applied, so the jets are longer than for the quantitative measurements explained in the text. The jets were longest (that is, most energetic) and contained the most atoms at values of τ_evolve for which the slope of the loss curve (Fig. 1b) was greatest. A tiny jet is barely visible for τ_evolve ≈ 2 ms (a), which is 1.7 ms before t_collapse. The images also show how the number of condensate atoms decreases with time. The time from the application of a_quench until the acquisition of the images was fixed at 5.2 ms.

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)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 τ_evolve are presented in Fig. 6. The jets exhibited variability in energy and number that was larger than the ∼10% measurement noise.

Figure 6: Quantitative jet measurements.

a, The number of atoms in the jets versus τ_evolve for the conditions of Fig. 1b. b, The spike density inferred from the kinetic energies of the jets. The bars indicate the range of shot-to-shot variability. For the analysis, we assumed the jets were disk-shaped, as the magnetic trap is axially symmetric. The images were taken perpendicular to the axial trap axis, viewing the disks edge-on. The jets expanded with velocity v ≈ 1 mm s^-1, which corresponds to a kinetic energy of ∼6 nK and a radial pre-quench gaussian r.m.s. width of ∼0.5 µm. As the axial size was below our resolution limit, we could not measure the axial expansion rate. For estimating the spike density, we assumed an axial width equal to the harmonic oscillator length. The atom density in the spikes decreased for larger values of |a_collapse|, and was half as large for a_collapse = -100a₀ as for -30a₀.

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 mean-field approach, and describe the condensate dynamics using the Gross–Pitaevskii (GP) equation. In most cases^12,13,15,16, 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 Ueda^15,16 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.

Theoretical challenges.

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 τ_decay is independent of both N₀ and |a_collapse| for |a_collapse| < 100a₀, 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 N₀ and a_collapse, but the fractions of remnant atoms, burst atoms, and missing atoms are independent of N₀.

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 high-energy atoms are being created, can we detect them? It is clear that adjustable interactions open up new avenues for BEC studies.