The dynamical behavior of Bose-Einstein condensation (BEC) in a gas with
attractive interactions is striking. Quantum theory predicts that BEC of a
spatially homogeneous gas with attractive interactions is precluded by a
conventional phase transition into either a liquid or solid. When confined to a
trap, however, such a condensate can form provided that its occupation number
does not exceed a limiting value. The stability limit is determined by a
balance between self-attraction and a repulsion arising from position-momentum
uncertainty under conditions of spatial confinement. Near the stability limit,
self-attraction can overwhelm the repulsion, causing the condensate to
collapse. Growth of the condensate, therefore, is punctuated by intermittent
collapses, which are triggered either by macroscopic quantum tunneling or
thermal fluctuation. Previous observation of growth and collapse has been
hampered by the stochastic nature of these mechanisms. Here we reduce the
stochasticity by controlling the initial number of condensate atoms using a
two-photon transition to a diatomic molecular state. This enables us to obtain
the first direct observation of the growth of a condensate with attractive
interactions and its subsequent collapse.Comment: 10 PDF pages, 5 figures (2 color), 19 references, to appear in Nature
  Dec. 7 200

CA Sackett

CC Bradley

Dmitry Strekalov

ERI Abraham

EV Shuryak

H-J Miesner

HT Stoof

Ionut Prodan

Jordan M. Gerton

M Houbiers

M Ueda

MR Andrews

PA Ruprecht

R Wynar

Randall G. Hulet

S Chandrasekhar

SL Cornish

Y Kagan

English

arXiv

Quantum theory predicts that Bose-Einstein condensation of a spatially homogeneous gas with attractive interactions is precluded by a conventional phase transition into either a liquid or solid [1]. When confined to a trap, however, such a condensate can form [2], provided that its occupation number does not exceed a limiting value [3, 4]. The stability limit is determined by a balance between the self-attractive forces and a repulsion that arises from position?momentum uncertainty under conditions of spatial confinement. Near the stability limit, self-attraction can overwhelm the repulsion, causing the condensate to collapse [5, 6, 7, 8]. Growth of the condensate is therefore punctuated by intermittent collapses [9, 10] that are triggered by either macroscopic quantum tunnelling or thermal fluctuation. Previous observations of growth and collapse dynamics have been hampered by the stochastic nature of these mechanisms. Here we report direct observations of the growth and subsequent collapse of a 7Li condensate with attractive interactions, using phase-contrast imaging. The success of the measurement lies in our ability to reduce the stochasticity in the dynamics by controlling the initial number of condensate atoms using a two-photon transition to a diatomic molecular state

Gerton, Jordan M.

Strekalov, Dmitry

Prodan, Ionut

Hulet, Randall G.

DSpace at Rice University

1Direct observation of growth and collapse of a Bose-Einsteincondensate with attractive interactionsJordan M. Gerton, Dmitry Strekalov*, Ionut Prodan, and Randall G. HuletPhysics and Astronomy Department and Rice Quantum Institute, MS 61, Rice University, Houston,Texas 77251, USA* Present address: Jet Propulsion Laboratory, MS 300-123, Pasadena, California 91109, USAThe dynamical behavior of Bose-Einstein condensation (BEC) in a gas with attractiveinteractions is striking.  Quantum theory predicts that BEC of a spatially homogeneous gas withattractive interactions is precluded by a conventional phase transition into either a liquid orsolid1.  When confined to a trap, however, such a condensate can form2 provided that itsoccupation number does not exceed a limiting value3,4.  The stability limit is determined by abalance between self-attraction and a repulsion arising from position-momentum uncertaintyunder conditions of spatial confinement.  Near the stability limit, self-attraction can overwhelmthe repulsion, causing the condensate to collapse5-8.  Growth of the condensate, therefore, ispunctuated by intermittent collapses9,10, which are triggered either by macroscopic quantumtunneling or thermal fluctuation.  Previous observation of growth and collapse has beenhampered by the stochastic nature of these mechanisms.  Here we reduce the stochasticity bycontrolling the initial number of condensate atoms using a two-photon transition to a diatomicmolecular state.  This enables us to obtain the first direct observation of the growth of acondensate with attractive interactions and its subsequent collapse.Condensate growth is initiated by cooling the gas below the critical temperature Tc for BEC.For attractive interactions, the number of condensate atoms N0 grows until the condensate collapses.During the collapse, the density rises giving a sharp increase in the rate of collisions, both elastic andinelastic.  These collisions cause atoms to be ejected from the condensate in an energetic explosion.The physics which determines both the stability condition and the dynamical process of collapse of thecondensate, draws some interesting comparisons to a star going supernova11, even though the time,length, and energy scales for these two phenomena are vastly different.  In the stellar case, the stabilitycriterion is provided by a balance between the pressure due to the quantum degeneracy of electronswhich make up the star and gravitational attraction.  If the mass of the star exceeds the stability limit12,the star collapses, releasing nuclear energy and triggering a violent explosion.  In contrast to the stellar2case, the condensate regrows after a collapse as it is fed by collisions between thermal atoms in the gas,and a series of sawtooth-like cycles of growth and collapse will continue until the gas reaches thermalequilibrium9,10.  We previously attained evidence for this nonequilibrium dynamical behavior in 7Liby measuring the distribution of N0 at selected times following a fast quench of the gas13.  N0 wasfound to be distributed between small numbers, N0 ≈ 100, and the maximum number of ~1250 atoms,in agreement with the growth and collapse model.The process of making reliable measurements of such small values of N0 destroys thecondensate, preventing an observation of condensate dynamics in real time.  Although the phase-contrast imaging technique employed here has been used for (nearly) nondestructive measurements oflarge condensates14, a few incoherent photons are always scattered, and these heat the gas.  As thephase-contrast signal is proportional to the number of scattered photons, achieving sufficient sensitivityto small condensates, such as those studied here, results in excessive heating and destruction of thecondensate15.  In the present work, direct observation of the dynamics is made possible by dumpingthe condensate, while only minimally affecting the thermal atoms.  This synchronizes the growth andcollapse cycles for different experiments at a particular point in time.  The subsequent growth/collapsedynamics are then obtained by repeating the experiment and measuring N0 at different delays followingthe dump pulse.  In a recent experiment with essentially pure condensates of 85Rb atoms, amagnetically-tuned Feshbach resonance was used to suddenly switch the interactions from repulsive toattractive and thereby induce a collapse at a specified time16.  In that experiment, condensates areproduced with N0 far greater than the stability limit, and consequently with high initial energy.  Incontrast, for the present experiment, the collapse occurs with N0 below the maximum number viamacroscopic quantum tunneling or thermal fluctuation7-9.  Furthermore, the condensate coexists with agas of thermal atoms, allowing the kinetics to be probed.The apparatus used to produce BEC of 7Li has been described previously15.  Permanentmagnets establish an Ioffe-Pritchard type trap with a nearly spherically symmetric, harmonic trappingpotential.  Approximately 2.5 × 108 atoms in the F = 2, mF = 2 hyperfine sublevel of 7Li are directlyloaded into the trap from a laser-slowed atomic beam. Following loading, the atoms are evaporativelycooled to ~400 nK with ~4 ×105 atoms using a microwave field to selectively spin-flip and remove themost energetic atoms.  Under these conditions, the gas is quantum degenerate.  The microwavefrequency is maintained at the end frequency for two seconds following evaporation, and is3subsequently lowered during a 100 ms duration quench pulse that removes all but the coldest ~105atoms.  This leaves the gas far from thermal equilibrium, and if left to freely evolve, the condensatewill alternately grow and collapse many times over a period of ~30 s13.  Destructive phase-contrastimaging is used to determine N0, the total number of atoms N, and their temperature T15.Following a delay of several seconds after the microwave quench pulse, the condensate isdumped by a light pulse consisting of two co-propagating laser beams whose frequency difference istuned to resonance between the collisional state of two free atoms and a vibrational level of thediatomic molecule Li2, as shown in Fig. 1.  Once in the molecular state, the laser of frequency ω2 canstimulate a single-photon transition to the intermediate level v′, which may spontaneously decay, mostlikely into a state of two energetic atoms that will escape the trap.  This method for removing atoms isvery energy specific since the observed two-photon linewidth of δ500 Hz is much less than the ~5 kHzthermal energy spread of the trapped atoms.  In particular, the condensate may be selectively removedwithout significantly affecting the remaining atoms.  This is demonstrated in Fig. 2, where both the EB∆2S + 2S2S + 2Pω2     ω1v = 10v'a3Σ+u13Σ+gEnergyInteratomic separationFigure 1  Two-photon molecular association technique.  Two atoms in the electronic groundstate (2S + 2S) interact along the +Σ ua3 diatomic molecular potential at the dissociation limitindicated by the dashed line.  They may be associated into the least-bound vibrational level, v =10, when the relative frequency between two drive lasers, ω1 - ω2, is equal to EB, where EB ≈12.47 GHz is the binding energy of the v = 10 level.  Each laser is tuned near resonance with avibrational level v′ = 72, of the +Σ g31 excited state potential.  The intensity of each laser beam isbetween 0.5 and 2.5 W/cm2 and the intermediate state detuning ∆ ≈ -110 MHz.4measured condensate fraction and the fraction of total atoms are plotted as a function of the duration ofthe light pulse.  The two-photon spectroscopy of this work is similar to that performed previously in anon-quantum-degenerate gas of lithium atoms at T ≈ 1 mK17, and in a Bose condensate of Rb atoms18.Following the light pulse, the gas is allowed to freely evolve for a certain time, at which point adestructive measurement of N0 is made.  Figure 3a shows the dynamical evolution of the condensatefollowing a light pulse whose duration is adjusted to reduce N0 to an initial value of ~100 atoms.  N0increases immediately following the light pulse as the condensate is fed via collisions betweennoncondensed thermal atoms, reaching a maximum value consistent with the expected upper limit of1250 atoms.  A collapse is clearly indicated by the subsequent reduction in N0.  After the collapse, N0grows again, since the gas is not yet in thermal equilibrium.  Figure 4 shows representative phase-contrast images taken from the data in Fig. 3a for the specified delay times.  The central peak, mostclearly visible at 450 ms delay, corresponds to the condensate.The condensate growth rate may be adjusted by varying the duration of the light pulse.  Byreducing the duration, fewer atoms are removed from both the condensate and from the low-energy0 5 10 15 20 25 30 35 400.00.51.0  Fraction remainingPulse duration (ms)Figure 2  Energy selectivity of the two-photon light pulse.  The fraction of condensate atoms(solid squares) and total atoms (open circles) which remain trapped immediately following alight pulse is plotted as a function of the light pulse duration.  Each data point is an average offive separate experiments normalized to measurements taken without the light pulse.  For thesemeasurements, N = (7 ± 1) ×104 atoms.  Achieving the necessary energy selectivity required thetwo diode lasers used to drive the two-photon transition to be phase-locked.  When locked, therelative frequency spectrum of the two lasers was measured to be less than 1 Hz in width.5thermal atoms that directly contribute to condensate growth, and consequently the growth rateincreases.  This is illustrated in Fig. 3b, where two secondary peaks are now discernible.  For Fig. 3c,the light pulse duration is lengthened, causing more atoms to be removed, and slowing the rate ofgrowth.03006009001,200    N003006009001,200ba   0.0 0.3 0.6 0.9 1.2 1.503006009001,200 c  Delay time (s)Figure 3  Condensate growth and collapse dynamics.  Each data point is the mean of 5measurements for the same delay time following the light pulse and the error bars are thestandard deviation of the mean.  a:  N0 is reduced to ~100 atoms using the two-photon lightpulse.  The reduction in N0 after the first peak at 450 ms is a direct manifestation of collapse.  b:The condensate is only partially dumped in order to speed up the initial growth.  Subsequentmaxima and minima are observed as the gas continues to undergo growth and collapse cycleswhile evolving towards thermal equilibrium.  c:  The light pulse duration is increased, so that thecondensate is dumped completely, to within the experimental resolution.  Note the slow turn onof growth and the subsequent saturation in the growth rate (see text).  For a and c, there is a 3 sdelay following the microwave quench pulse before the light pulse, while in b, the delay is 5 s.Additionally, for a and c, N = (7 ± 1) ×104 atoms and T = (170 ± 15) nK immediately before thelight pulse, while for b, N = (1.0 ± 0.1) ×105 atoms and T = (200 ± 20) nK.  For each individualimage, the statistical uncertainty in N0 is ±60 atoms, while the systematic uncertainty, dominatedby uncertainties in the imaging system, is ±20%13.6Each data point in Fig. 3 is the mean of five separate measurements of N0.  Consequently, thesedata represent an average of many trajectories whose initial phase and rate of growth differ slightly.  Toanalyze the results, we numerically simulate the collisional redistribution of atoms over the energyFigure 4  Phase-contrast images.  These images were selected from those used to constructFig. 3a.  The upper image corresponds to data immediately after the dump pulse (0 ms), themiddle to the peak of condensate growth (450 ms), and the lower to the first collapse (550 ms).The fitted values of N0 are 40, 1210 and 230 atoms for the upper, middle and lower images,respectively.  The displayed images result from angle-averaging the actual images about theprobe laser propagation axis and the signal height is proportional to the column density of theatom cloud integrated along this direction4.  The image area corresponds to an 85 µm square.A movie of the growth and collapse was produced using representative time-ordered imagesfrom the data of Fig. 3a, and is available for viewing at http://atomcool.rice.edu/collapse.html.7states of the trap using the quantum Boltzmann equation, as described in detail elsewhere9.  Thecolored curves shown in Fig. 5 are a sample of simulated trajectories which include the effect of themicrowave quench pulse and the light pulse. The variation in condensate growth following a light pulseis mainly the result of slight differences in initial conditions which lead to variations in the energydistribution of atoms in the trap.  Additionally, the stochastic nature of the collapse process, whichcauses each collapse to occur at a slightly different value of N0, contributes to dephasing of differenttrajectories.  The heavy black line represents the average of 40 trajectories obtained by multiplyrunning the simulation with slightly different initial conditions.  In the simulations, N0 is set to 100atoms immediately following the light pulse in order to provide direct comparison with the data in Fig.3a. Additionally, N0 is set to 200 atoms immediately following a collapse in order to achieve the bestagreement with our previous statistical studies13.  The only adjustable parameter in the simulationswas the fraction of thermal atoms lost within the spectral width of the two-photon transition.The simulation results agree well with the data in several respects.  The data in Fig. 3b showthat each subsequent peak following the initial growth is slightly lower, as the trajectoriescorresponding to each individual measurement dephase from one another.  This dephasing causes the0.0 0.3 0.6 0.9 1.2 1.503006009001,200  N0Delay time (s)Figure 5  Simulation of condensate dynamics.  The colored curves show individual trajectoriesgenerated by a numerical quantum Boltzmann simulation, which models the kinetics ofevaporative cooling and thermal equilibration.  Individual trajectories were generated by runningthe simulation for 40 different initial values of N spaced evenly between 2.4×108 and 2.6×108atoms.  The resulting values of N just before the light pulse range between 6×104 and 8×104atoms, in agreement with the range of N observed in the data.  The heavy solid line is the meanof these 40 trajectories.  The effect of the light pulse on the thermal atoms is simulated byinstantaneously reducing the population of atoms within a 500 Hz wide Lorentzian energy band,consistent with the observed spectral width.  For the simulation shown, 90% of the thermalatoms at the Lorentzian peak are lost.8maxima (minima) to occur at smaller (larger) values of N0 than for any individual trajectory.  Thisbehavior is seen in the simulation average shown in Fig. 5, confirming our understanding of the role ofaveraging in these measurements.Condensate growth should be affected by the quantum statistical effect known as Boseenhancement: the condensate growth rate scales with the occupation number N0.  This would lead to anexponentially increasing growth rate, which is, however, neither observed in the data nor predicted bythe quantum Boltzmann equation model.  For the conditions shown in Fig. 3c, for example, both themodel and the data show a saturation in the growth rate when N0 becomes larger than ~150 atoms.  Ananalysis of the distribution of atoms among the trap energy levels in the model simulations indicatesthat the condensate is fed mostly by collisions between atoms with the lowest energies.  Further, thisanalysis indicates that when the low-energy population is depleted, the growth is limited by the rate fornon-Bose-enhanced ‘trickle-down’ collisions between high-energy atoms.  Due to our high sensitivityto small values of N0, the very early stages of condensate growth following the dump can be observed,illuminating this subtle, yet important, departure from the pure Bose enhancement prediction.  Thework of Miesner et al.19 was the first observation of condensate growth, but that experiment could notdetect small values of N0 and therefore was not sensitive to the initial stages of growth.The data presented here are the first direct observations of the growth and collapse of Bose-Einstein condensates with attractive interactions.  By enabling the preparation of condensates with adesired mean number of atoms with minimal perturbation to the noncondensed atoms, the two-photontechnique provides a new way of studying the dynamics of this non-equilibrium and stochasticquantum system.Acknowledgements.  We thank Cass Sackett for help with the quantum Boltzmann simulation.  This work was supportedby the U.S. National Science Foundation, the National Aeronautics and Space Administration, the Office of NavalResearch, and the Welch Foundation.9References1. Stoof, H. T. C. Atomic Bose gas with a negative scattering length. Phys. Rev. A 49, 3824-3830(1994).2. Bradley, C. C., Sackett, C. A., Tollett, J. J. & Hulet, R. G. Evidence of Bose-Einsteincondensation in an atomic gas with attractive interactions. Phys. Rev. Lett. 75, 1687-1690(1995).3. Ruprecht, P. A., Holland, M. J., Burnett, K. & Edwards, M. Time-dependent solution of thenonlinear Schrodinger equation for Bose-condensed trapped neutral atoms. Phys. Rev. A 51,4704-4711 (1995).4. Bradley, C. C., Sackett, C. A. & Hulet, R. G. Bose-Einstein condensation of lithium:observation of limited condensate number. Phys. Rev. Lett. 78, 985-989 (1997).5. Kagan, Y., Shlyapnikov, G. V. & Walraven, J. T. M. Bose-Einstein condensation in trappedatomic gases. Phys. Rev. Lett. 76, 2670-2673 (1996).6. Shuryak, E. V. Bose condensate made of atoms with attractive interaction is metastable. Phys.Rev. A 54, 3151-3154 (1996).7. Stoof, H. T. C. Macroscopic quantum tunneling of a Bose condensate. J. of Stat. Phys. 87,1353-1366 (1997).8. Ueda, M. & Leggett, A. J. Macroscopic quantum tunneling of a Bose-Einstein condensate withattractive interactions. Phys. Rev. Lett. 80, 1576-1579 (1998).9. Sackett, C. A., Stoof, H. T. C. & Hulet, R. G. Growth and collapse of a Bose condensate withattractive interactions. Phys. Rev. Lett. 80, 2031-2034 (1998).10. Kagan, Y., Muryshev, A. E. & Shlyapnikov, G. V. Collapse and Bose-Einstein condensation ina trapped Bose gas with negative scattering length. Phys. Rev. Lett. 81, 933-937 (1998).11. Houbiers, M. & Stoof, H. T. C. Stability of Bose condensed atomic 7Li. Phys. Rev. A 54, 5055(1996).1012. Chandrasekhar, S. An Introduction to the Study of Stellar Structure (Dover, New York, 1957).13. Sackett, C. A., Gerton, J. M., Welling, M. & Hulet, R. G. Measurements of collective collapsein a Bose-Einstein condensate with attractive interactions. Phys. Rev. Lett. 82, 876-879 (1999).14. Andrews, M. R. et al. Direct, nondestructive observation of a Bose condensate. Science 273, 84-87 (1996).15. Sackett, C. A., Bradley, C. C., Welling, M. & Hulet, R. G. Bose-Einstein condensation oflithium. Appl. Phys. B 65, 433-440 (1997).16. Cornish, S. L., Claussen, N. R., Roberts, J. L., Cornell, E. A. & Wieman, C. E. Stable 85RbBose-Einstein condensates with widely tunable interactions. Phys. Rev. Lett. 85, 1795-1798(2000).17. Abraham, E. R. I., McAlexander, W. I., Sackett, C. A. & Hulet, R. G. Spectroscopicdetermination of the s-wave scattering length of lithium. Phys. Rev. Lett. 74, 1315-1318 (1995).18. Wynar, R., Freeland, R. S., Han, D. J., Ryu, C. & Heinzen, D. J. Molecules in a Bose-Einsteincondensate. Science 287, 1016-1019 (2000).19. Miesner, H.-J. et al. Bosonic stimulation in the formation of a Bose-Einstein condensate.Science 279, 1005-1007 (1998).

Direct observation of growth and collapse of a Bose-Einstein condensate with attractive interactions

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Direct observation of growth and collapse of a Bose-Einstein condensate with attractive interactions

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