Trapping, tunneling & fragmentation of condensates in optical traps

Transcription

1 Trapping, tunneling & fragmentation of condensates in optical traps Nimrod Moiseyev Department of Chemistry and Minerva Center for Non-linear Physics of Complex Systems, Technion-Israel Institute of Technology.

2 N. Moiseyev, L. D. Carr, B. A. Malomed, and Y. B. Band, Transition from resonances to bound states in nonlinear systems: Application to Bose-Einstein condensates. Journal of Physics B, (004), 37(9), L193-L00. L.S. Cederbaum and N. Moiseyev, On the collapse and restoration of condensates in n dimensions in the mean-field approximation. Israel Journal of Chemistry (003), 43(3-4), N. Moiseyev and L. S. Cederbaum, Tunneling lifetime of trapped condensates. Los Alamos National Laboratory, Preprint Archive, Condensed Matter (004), 1-11, arxiv:cond-mat/

3 Resonances Are localized metastable states with finite lifetime. In Hermitian Quantum Mechanics resonances cannot be represented by a single state of the Hamiltonian. The resonance is depicted by a large density of states around the resonance energy. Resonances are associated with the complex poles of the scattering matrix.

4 E E Hermitian QM V( x) Ae ikx + Be ikx ( k) = h continuous E > 0 spectrum m = ε ± Γ res 1 E E Non Hermitian QM V( x) Ae ikx + Be ikx i = εn Γ discrete spectrum n resonances i = ε Γ i k = k e ϕ 1 1 E-between resonances

5 Hermitian QM Non Hermitian QM V( x) V( x) Complex scaling Ae ikx + Be ikx Ae ikxe iθ + Be ikxe iθ E ( k) = h continuous E > 0 spectrum m E i = εn Γ n discrete spectrum resonances E = ε ± Γ res 1 E-between resonances E i = ε Γ i k = k e ϕ k = 1 1 i i k = k e ϕ e θ continuum: i k e θ Ae to avoid divergence- ikxe iθ + Be ikxe iθ i E = E e θ

6 Complex Scaling In order to avoid the divergence of the resonance wavefunction it is convenient to scale the coordinate such that : x i xe θ This can be done by the following scaling operators: i x Sˆ e θ x Such that ˆ Sψ 0 The Schrodinger equation takes the form: ˆ ˆ ˆ 1 ~ ( S HS )( Sˆ ψ ) = E( Sˆ ψ ) H ~ ψ = E ~ ψ res θ as θ θ θ

7 Complex scaling: Reviews W. P. Reinhardt, Annu. Rev. Phys. Chem. 33, 3 (198) N. Moiseyev. Phys. Rep. 30, 11 (1998) Reflection free CAPs by the Smooth-Exterior-Scaling transformation N. Moiseyev, J. Phys. B, 31, 1431, (1998)

8 Hermitian (conventional ) QM variational calculations (numerical exact) Bound, resonance, continuum states for 1-atom in 1D trap

9 Non-hermitian QM variational calculations Bound, tunneling resonance, continuum and above-barrier resonance states

10 Bound state Resonances E j i = ε j Γ j Continua Non-interacting atoms in 1D optical trap (odd-parity resonances obtained in 3D spherical symmetric potential)

11 BEC-model Hamiltonian GP: N N r 0 ( ) a r r T + V j + δ ( j j' ) ψ = εψ j= 1 j' j r r ψ = φ ( )... φ ( ) r1 r N U = a ( N 1) 0 r U r ε U r 4 r T + V( ) + φ( ) φ = µφ E = = µ () φ dr N if N > N ( GP : U > U ) c c a 0 > 0 a 0 < 0 Bound to resonance state transition Resonance to bound state transition

12 OPEN QUESTIONS How resonances can be calculated for the NLSE (GP)? In 3D, negative scattering length BEC: are the resonance/bound-state transitions take place before the collapse of the BEC? How the fraction of the atoms that are tunneled through the external potential barriers can be extracted from the GP calculations?

13 How resonances can be calculated for the NLSE (GP)? A: Complex scaling: r r r r r r r q = re Tθ = e T Vθ = V( re ) δ( q q ) = e δ( ) + iθ iθ iθ iθn r + j j j j j' j j' j j j NLSE ( GP ): * Hθ = Hθ φθ = φθ r φθ = φθ r () () iθ r + iθ U iθ r r r e Tr + V( re ) + e φθ ( ) φ θ( ) = µ ( complex) φθ( ) γ U = a ( N 1) 0 1 ( N) = Im γ / h = dn N dt

14 Resonances for BEC with a positive scattering length ε U r 4 r E = = µ φ( ) dr N γ = Im µ Γ = Im E

15 Q: Why Ec< 0? A(?): The threshold for 1-atom (chemical pot) is 0. The threshold of E <0 is due to fraction of N atoms that tunnel through the potential barriers Energy and Chemical potential Transition from bound (U<Uc) to resonance (U>Uc)

16 How resonances can be calculated for the NLSE (GP)? B: Smooth-Exterior Complex Scaling (SES): r r r r r r r r q = F ( ) T = T + V V( ) = V( F ( )) δ ( q q ) = δ ( F ( ) F ( )) r j θ j θ SES CAP j θ j j' j θ j' θ j j d V V x V x V x dx d dx θ θ θ SES CAP = 0 ( ) + 1 ( ) + ( ) r r r r Tr + V( ) + UφSES ( ) + VSES CAP φ SES ( ) = µ ( complex) φses ( ) r r φ ( ) φ ( ) ( ) r SES SES NLSE ( GP ): Assumption: the atoms tunneling outside do not interact. δ ( q r q r ) = δ ( r r ) j ' j j' j (out going flux in GP Eq. )

17 Resonance to bound-state transitions for BEC with negative scattering length ( SES-CAP approximated by a local CAP)

18 1D BEC 1 d U = dx V ( x) φcap ( x) VCAP φ CAP ( x) µ ( complex) φcap ( x) U U = a ( N 1) < 0 0 0

19 D BEC U Tρ + V ρ + φ ρ + V ρ φ ρ = µφ ρ U U = a N 4πρ ( ) CAP( ) CAP( ) CAP( ) CAP( ) ; 0 0( 1)

20 3D BEC U T V r r V r r r U U a N 8π r r + () + φ () () () () ; CAP + CAP φ CAP = µφcap 0 = 0( 1) < 0 Collapse before resonance/bound Transitions!

21 In 3D only the odd states survive In 3D NO BOUND STATE In 3D Only ONE resonance tunneling state survives

22 3D optical trap Resonance to bound-state transition BEFORE collapse of BEC

23 Q: Can we avoid the collapse in 3D BEC? in spherical symmetric optical trap The key point is to associate the collapse phenomena with the unbounded spectrum from below due to the - U /r^ term which acts like a black hole Therefore, V(r) should be LESS singular than 1/r^ at the origion The excluded volume idea: Vr () = when r< r 0 With Cederbaum we proved for Harmonic potential: U U φ ( r) dr J ( r) φ ( r) dr where 0 J ( r) 1 r = 4 4 HO 3 0 HO 3 0 r0 NO COLLAPSE

24 Q: Why Ec< 0? A(?): The threshold for 1-atom (chemical pot) is 0. The threshold of E <0 is due to fraction of N atoms that tunnel through the potential barriers AS WE HAVE SHOWN BEFORE Energy and Chemical potential Transition from bound (U<Uc) to resonance (U>Uc)

25 1 Q: Why Ec<0? A: due to fragmentation µ = µ ( U) = 0 E = EU ( ) < 0 c c c c n atoms in the potential well occupy the φ orbital n atoms tunnel through the potential barriers occupy the χ orbital Fraction remains inside the trap X = a a 0 0 U ( N 1) N n 1 N n 1 X = N = E ( XU, ) = XE( XU) + (1 X) E [(1 XU ) ] BEC Pitaevskii (Re v. Mod. Phys. 1999) E χ = 0 E (, ) ( ) BEC X U = XE XU χ

26 Transition from bound to a resonance state (positive scattering length) occurs at: Uc=0.879 Fraction of atoms inside the potential well X Bound/resonance transition Xc(U)=Uc/U The bound/resonance transition should happen at the minimal energy where: E BEC X X c ( U) = 0

27 The bound/resonance transition should happen at the minimal energy where: E BEC X X c ( U) = 0

28 How resonances can be calculated for the NLSE (GP)? A: By using complex scaling with inversed complex scaled scattering length, or by introducing the reflection-free CAPs derived from the smooth-exterior-scaling procedure. In 3D BEC (negative scattering length): are the resonance/bound-state transition take place before the collapse of the BEC? A: Construct an optical trap with a resonance state close to the threshold OR a spherically symmetric trap with excluded volume to prevent the atoms to approach the central point r=0. How the fraction of the atoms that are tunneled through the external potential barriers can be extracted from the GP calculations? A: Define E(X,U)=XE(X,U) where E the mean energy per atom obtained in GP calculations where, X=(number of atoms remain inside the trap)/n Resonance to bound state transitions occur at Xc, E ( X, U ) X Xc = 0

29 Γ= ImE is function of N Rate of decay of N atoms in the well to n_1<n atoms where (N-n_1) tunnel through the poetntail barriers.

30 N. Moiseyev, L. D. Carr, B. A. Malomed, and Y. B. Band, Transition from resonances to bound states in nonlinear systems: Application to Bose-Einstein condensates. Journal of Physics B, (004), 37(9), L193-L00. L.S. Cederbaum and N. Moiseyev, On the collapse and restoration of condensates in n dimensions in the mean-field approximation. Israel Journal of Chemistry (003), 43(3-4), N. Moiseyev and L. S. Cederbaum, Tunneling lifetime of trapped condensates. Los Alamos National Laboratory, Preprint Archive, Condensed Matter (004), 1-11, arxiv:cond-mat/