Collapse dynamics of a 176 Yb - 174 Yb Bose-Einstein condensate
aa r X i v : . [ c ond - m a t . qu a n t - g a s ] J un Collapse dynamics of a
Yb -
Yb Bose-Einstein condensate
G. K. Chaudhary ∗ and R. Ramakumar † Department of Physics and Astrophysics, University of Delhi, Delhi-110007, India (Dated: 4 June 2010)In this paper, we present a theoretical study of a two-component Bose-Einstein condensate com-posed of Ytterbium (Yb) isotopes in a three dimensional anisotropic harmonic potential. Thecondensate consists of a mixture of
Yb atoms which have a negative s-wave scattering lengthand
Yb atoms having a positive s-wave scattering length. We study the ground state as well asdynamic properties of this two-component condensate. Due to the attractive interactions between
Yb atoms, the condensate of
Yb undergo a collapse when the particle number exceed a criticalvalue. The critical number and the collapse dynamics are modified due to the presence of
Ybatoms. We use coupled two-component Gross-Pitaevskii equations to study the collapse dynam-ics. The theoretical results obtained are in reasonable agreement with the experimental results ofFukuhara et al. [PRA , 021601(R) (2009)]. PACS numbers: 03.75.Hh, 03.75.Kk, 03.75.Mn
I. INTRODUCTION
The first experimental observation of Bose-Einstein condensate (BEC) in bose atom vapors have initiated anexciting field of research, both theoretically and experimentally. One of the most interesting developments in thisfield is the formation of multi-component condensates. Multi-component BECs have been observed experimentallyby Myatt et al. and Hall et al. for two different hyperfine spin sates of Rb, by Modugno et al. for differentatoms ( K and Rb), and Papp et al. for different isotopes of the same atom ( Rb and Rb). A rich varietyof various interesting effects exhibited by these two-component BECs have inspired a number of theoretical studiescovering various aspects of a these systems . The common feature of the bose systems in these experiments isthat the intra-component and the inter-component boson-boson interactions are all repulsive. It rises the curiosityabout the properties of a multi-component condensate in which one kind of atoms have repulsive interactions whileanother kind of atoms have attractive interactions. Recently, Fukuhara et al. observed BEC of spin-zero Yb isotopesby implementing an all-optical cooling protocol. The bose-bose mixture in these experiments contain Yb atomshaving a positive s-wave scattering length and
Yb atoms having a negative s-wave scattering length . The s-wave scattering length between
Yb and
Yb is also positive . Such a two-component condensate can be expectedto show dynamical properties far more complex than a one-component condensate of attractively interacting bosons,which becomes unstable when the number of atoms exceed a critical value .For a two-component BEC with attractive interactions between bosons in one component and repulsive interactionsin the second component, two most basic questions are of that of its stability and that of the collapse dynamics. In thispaper, we theoretically study the ground state as well as dynamic properties of such a two-component condensate ina three dimensional (3D) anisotropic harmonic potential. We specifically consider the Yb-
Yb bose-bose mixturein the anisotropic harmonic confining potential as in the experiment of Fukuhara et al. . The paper is arranged asfollows. In Sec. II, we describe the theoretical model for the study of a two-component BEC. In Sec. III, we discussthe ground state profile of a stable two-component BEC. In Sec. IV, we present the collapse dynamics of the systemand compare it with the experimental results . The conclusions are given in Sec. V. II. TWO-COMPONENT BEC: THEORETICAL MODEL
The ground state and dynamic properties of a two component Bose-Einstein condensate (BEC)is well described bya set of coupled Gross-Pitaeveskii equations (GPE) given by i ~ ∂ψ i ( r , t ) ∂t = ( − ~ m i ∇ + v i ( r ) + g ii | ψ i ( r , t ) | + g ij | ψ j ( r , t ) | ) ψ i ( r , t ) , (1)where i = 1 , Yb and 2 for
Yb), j = 3 − i , r ≡ ( x, y, z ) T is the spatialcoordinate vector, v i ( r ) = (1 / m i ( ω x x + ω y y + ω z z ) is the trapping potential, g ii = 4 π ~ a ii /m i is the intra-speciesinteraction and g ij = 2 π ~ a ij /m ij the inter-species interaction strength between atoms in the condensed state, a ii isintra-species and a ij is inter-species s-wave scattering length, m ij = m i m j / ( m i + m j ) is the reduced mass in which -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 |ψ (0,y,z) | y z |ψ (0,y,z) | -4 -3 -2 -1 0 1 2 3 4 -4 -3 -2 -1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 |ψ (0,y,z) | y z |ψ (0,y,z) | FIG. 1: Ground state profiles of
Yb (top panel) and
Yb (bottom panel) condensates in a 3d anisotropic harmonic potentialwith N = 6 × and N = 150. m i and m j are atomic masses. The normalization condition for each component is R | ψ i ( r ) | dr = N i , where N i isnumber of atoms in each component.We non-dimensionalize Eq. (1) through a set of linear transformations: ˜ t = ω x t, ˜ r = r /l, e ψ i ( r ) = N − i l / ψ ( r ).After dropping the wiggles on the symbols, we obtain i ∂ψ i ( r , t ) ∂t = ( − ∇ + v i ( r ) + λ ii | ψ i ( r , t ) | + λ ij | ψ j ( r , t ) | ) ψ i ( r , t ) , (2)where l = r ~ mω x , λ ii = 4 πa ii l , λ ij = 4 πa ij l ,v i ( r ) = 12 ( x + κ y + γ z ) , κ = ω y ω x , γ = ω z ω x . (3)Since the masses of the two isotopes are nearly equal, we have taken m = m = m . In order to find a stationarysolution of Eq. (2), we do a separation of variables ψ i ( r , t ) = ψ i ( r ) × exp[ − i ( µ i / ( ~ ω x )) t ], where µ i is the chemicalpotential of the i th component. Starting from Eq. (2), we obtain( − ∇ + v i ( r ) + λ ii | ψ i ( r ) | + λ ij | ψ j ( r ) | ) ψ i ( r ) = µ i mω x ψ i ( r ) . (4) III. GROUND STATE PROFILES OF A TWO-COMPONENT BEC OF YB ATOMS
In this section, we discuss the ground state properties of a two-component BEC of Yb isotopes by numericallysolving the coupled GPE (Eq. 4). The ground state solution of the GPE is found by the imaginary time propagationmethod. In this method, the time dependent GPE is evolved in imaginary time starting from an initial guess usinga finite difference Crank-Nicholson (FDCN) scheme . In imaginary time propagation we have taken the space stepas δx = δy = δz = 0 . δt = 0 . Yb -
Yb system in the experiment : m = 2 . × − Kg , a = 5 . × − m , a = − . × − m , a = a = 2 . × − m , ν x (= ω x / π ) = 45 Hz , ν y (= ω y / π ) = 200 Hz , ν z (= ω z / π ) = 300 Hz .Due to the attractive interaction between Yb atoms, the condensate of
Yb undergo a collapse if the particlenumber exceeds a critical value N c . For a single component Yb this value is given by N c ≈ . L/ | a | , where N , N t N , N t N , N t N , N t N , N t N , N t FIG. 2: Time evolution of number of
Yb(filled circles),
Yb (open squares, open circles, filled squares). The correspondingexperimental results are given by (open triangles for
Yb and filled triangles for
Yb). The values of three body recombina-tion terms are: K = 4 . × − cm s − (for all points)), K = 3 . × − cm s − (open squares), K = 3 . × − cm s − (opencircles), K = 3 . × − cm s − (filled squares). The time is in units of seconds. L = p ~ /mω and ω = ( ω x ω y ωz ) . For the parameters given above, N c ≈ Yb atoms with a positive scattering length. The physical origin of this modification is inthe effective potential felt by the bosons in the attractive component. The interaction with the repulsive componentchanges the effective potential from v ( r ) − | λ || ψ ( r ) | to v ( r ) − | λ || ψ ( r ) | + λ | ψ ( r ) | . This reduces theeffect of the attractive interaction and leads to a reduction of the critical number. Alternatively, one may say thatthe mean-field contribution from the repulsive component leads to a flattening of the effective potential felt by theattractive component. If N = 6 × , then for the given parameters, N c is calculated to be ≃
220 by our numericalsimulation. In order to prepare a stable condensate, we have taken N = 6 × and N = 150. The ground sateprofile of the two-component BEC is presented in Fig. 1. We observe that the Yb atoms are at the center of trapand are surrounded by a large cloud of
Yb atoms.
IV. COLLAPSE DYNAMICS OF A TWO-COMPONENT BEC OF YB ATOMS
In this section, we study the collapse dynamics of a two component BEC composed of
Yb and
Yb atoms usingthe coupled time-dependent GPE’s. Due to the negative s-wave scattering length, the
Yb condensate becomesunstable if the number of atoms becomes greater than a critical value N c . This condensate collapses by emittingatoms out of it. Due to high density of atoms in the attractive condensate, the loss of atoms from the condensateoccurs through three-body collisions. To model this collapse we add a imaginary three body quintic loss term tothe RHS of GPE Eq. (1) given by K i ψ i = − (1 / iK i | ψ i | ψ i , (5)where K i is the three-body loss coefficient for each component. We have neglected the two-body dipolar loss term asthey make negligible contribution in this case . We have also left out the loss terms for Yb − Yb − Yb, Yb − Yb − Yb, Yb − Yb − Yb, and Yb − Yb − Yb collisions, since the losses due to theseare comparatively small . So, Eq. (2) becomes i ∂ψ i ( r , t ) ∂t = ( − ∇ + v i ( r ) + λ ii | ψ i ( r , t ) | + λ ij | ψ j ( r , t ) | − iξ i | ψ i ( r , t ) | ) ψ i ( r , t ) , (6)where ξ i = (1 / N K i l − ω − x . We time evolve the coupled time-dependent GP equations Eq. (6) using the finite difference Crank-Nicholson(FDCN) scheme with a known initial condition. In real time propagation, we have taken the space step as δx = δy = δz = 0 . δt = 0 . Yb and
Yb isshown in Fig. 2 along with the experimental data of Fukuhara et al. . Considering the complex dynamics of the -4 -3 -2 -1 0 1 2 3 4-4-3-2-1 0 1 2 3 4 0 0.2 0.4 0.6 0.8 1 1.2 1.4 1.6 |ψ (0,y,z) | y z |ψ (0,y,z) | -4 -3 -2 -1 0 1 2 3 4-4-3-2-1 0 1 2 3 4 0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1 |ψ (0,y,z) | y z |ψ (0,y,z) | -4 -3 -2 -1 0 1 2 3 4-4-3-2-1 0 1 2 3 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 |ψ (0,y,z) | y z |ψ (0,y,z) | -4 -3 -2 -1 0 1 2 3 4-4-3-2-1 0 1 2 3 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 0.18 |ψ (0,y,z) | y z |ψ (0,y,z) | FIG. 3: Ground state profile of
Yb at different times. The four figures correspond to t ( seconds ) = 0 . . . K = 4 . × − cm s − , K = 3 . × − cm s − . Here, N = 6 × and N = 2 × . the two-component system with mixed interactions, the theoretical results may be said to be in reasonable agreementwith the experimental results. We observe that there is a significant loss of Yb atoms. We also see that the decayof
Yb is very rapid. It is due to the collapse of the
Yb condensate. The number of
Yb atoms does show avery small decrease, which is not visible on the scale of this figure. To understand the details of the decay process,we study the condensate profiles of each component at different times. The results are shown in the Fig. 3 for
Yband in Fig. 4 for
Yb. At t = 0, the Yb are at the center of the trap (top left panel of Fig. 3) surrounded by
Yb (top left panel of Fig. 4). Since the number of atoms in the attractive component is higher than the criticalnumber for stability, the system is unstable. When the system evolves in time, the attractive component explodes asis evident from the spreading of this component with time, in real space, as shown in Fig. 3. The spiky structures inthese figures represent the inhomogeneities produced due to the on-going explosion process. As mentioned earlier, theexplosion also leads to a spread of the ground state profile. Due to the coupling between the attractive and repulsivecomponents, the condensate of
Yb is also redistributed in real space during the time evolution, as shown in Fig.4. The numbers of remaining atoms in each condensate component during the time evolution, for a longer period oftime, is shown in Fig. 5. We note that the best agreement with the experimental results (Fig. 2) is obtained for themeasured values of the K and K . The disagreement at later times is likely to be originating from the neglect ofatomic loss due to collisions involving Yb and
Yb. During the initial stages of the time evolution, the bosonsdistribution gets heavily mixed due to the explosion and due to the coupling between the two components. Then, atlater times, the inter-component collisions is likely to affect the atom loss. We are unable to include these loss termssince their values are not known at present.
V. CONCLUSIONS
In this paper, we presented a study of some static and dynamic properties of a two-component bose condensateconsisting of repulsively interacting
Yb atoms and attractively interacting
Yb atoms in an anisotropic harmonicconfinement. In the stable state, the ground state has
Yb atoms in the center of the trap surrounded by
Ybatoms. When the number of atoms in the attractive component exceed a critical value, the the system undergo acollapse. We analyzed the time evolution of this collapse process for the specific system parameters of the
Yb -
Yb system studied in the experiment of Fukuhara et al. . The details of the collapse dynamics are found to bein reasonable agreement with experimental results. The critical number for stability of the attractive condensate isreduced by it’s interaction with the repulsive condensate. -4 -3 -2 -1 0 1 2 3 4-4-3-2-1 0 1 2 3 4 0 0.02 0.04 0.06 0.08 0.1 0.12 0.14 0.16 |ψ (0,y,z) | y z |ψ (0,y,z) | -4 -3 -2 -1 0 1 2 3 4-4-3-2-1 0 1 2 3 4 0 0.02 0.04 0.06 0.08 0.1 0.12 |ψ (0,y,z) | y z |ψ (0,y,z) | -4 -3 -2 -1 0 1 2 3 4-4-3-2-1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 |ψ (0,y,z) | y z |ψ (0,y,z) | -4 -3 -2 -1 0 1 2 3 4-4-3-2-1 0 1 2 3 4 0 0.05 0.1 0.15 0.2 0.25 |ψ (0,y,z) | y z |ψ (0,y,z) | FIG. 4: Ground state profile of
Yb at different times. The four figures correspond to t ( seconds ) = 0 . . . K = 4 . × − cm s − , K = 3 . × − cm s − . Here, N = 6 × and N = 2 × .
100 1000 10000 100000 0 0.5 1 1.5 2 N , N t
100 1000 10000 100000 0 0.5 1 1.5 2 N , N t
100 1000 10000 100000 0 0.5 1 1.5 2 N , N t
100 1000 10000 100000 0 0.5 1 1.5 2 N , N t
100 1000 10000 100000 0 0.5 1 1.5 2 N , N t
100 1000 10000 100000 0 0.5 1 1.5 2 N , N t FIG. 5: Time evolution of number of
Yb(top curve),
Yb(solid line, dashed, dash-dot lines). The values of three bodyrecombination terms are: K = 4 . × − cm s − (for all lines)), K = 3 . × − cm s − (solid line), K = 3 . × − cm s − (dashed line), K = 3 . × − cm s − (dash-dot line). The time is in units of seconds. VI. ACKNOWLEDGMENTS
Gopesh Kumar Chaudhary thanks UGC, Government of India, for financial support during this work. GKC thanksProf. G. V. Shlyapnikov for a discussion during the ICTS International School on Cold Atoms and Ions held recentlyin Kolkata. We thank Prof. J. K. Bhattacharjee for a discussion during an SERC school on Nonlinear Dynamics heldrecently in Delhi. ∗ Electronic address: [email protected] † Electronic address: [email protected] M. H. Anderson, J. R. Ensher, M. R. Matthews, C. E. Weiman, and E. A. Cornell, Science , 198 (1995). K. B. Davis, M. -O. Mewes, M. R. Andrews, N. J. van Druten, D. S. Durfee, D. M. Kurn, and W. Ketterle, Phys. Rev. Lett. , 3969 (1995). C. C. Bradley, C. A. Sackett, J. J. Tollett, and R. G. Hulet, Phys. Rev. Lett. , 1687 (1995). C. J. Myatt, E. A. Burt, R. W. Ghrist, E. A. Cornell, and C. E. Wieman, Phys. Rev. Lett.
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