Inclusion of Mean Field Effects in the Two-Mode Model of BECs in a Double Well David Masiello in collaboration with Bill Reinhardt University of Washington, Seattle August 8th, 2005 masiello@u.washington.edu
Outline Part 1 (What Bill would ve talked about) BEC Fragmentation MOs and molecular dissociation Energy level correlation diagrams Oscillator, fragmented, & cat states Analog of Anderson s pendulum Generating & detecting le chat de Schrödinger
Outline Continued Part 2 Build in effects of mean field Hartree-Fock for bosons Confusing role of factor of 2 in Hartree- Fock contact interaction Energy level correlation diagrams Oscillator, fragmented, and cat states Lively discussion
BEC Double Well Problem External potential V (x) Bose-Hubbard description 2 modes χ L (x) χ R (x) χ A (x) GP or mean field description χ S (x) 1 mode
Why are Correlations Important in Fragmentation? Chemical problem: Molecular dissociation H 2 H + H Molecular orbital theory works well near equilibrium ground state geometry of Ψ MO (x 1, x 2 ) = [χ L (x 1 ) + χ R (x 1 )][χ L (x 2 ) + χ R (x 2 )] This superposition is a GP type ansatz H 2
R 1 R 2 But as... Molecular orbital theory fails Ψ MO (1, 2) = [χ L (1) + χ R (1)][χ L (2) + χ R (2)] = χ L (1)χ L (2) + χ R (1)χ R (2)+χ L (1)χ R (2) + χ R (1)χ L (2) Description of dissociated H atoms needs two configurations Ψ CI (1, 2) = C 1 [χ L (1) + χ R (1)][χ L (2) + χ R (2)] +C 2 [χ L (1) χ R (1)][χ L (2) χ R (2)] GP theory, being one configuration, cannot do this!
Spekkens-Sipe Two-Mode Model (PRA, 1999) Hamiltonian with V (x x ) = gδ(x x ) Ĥ = { d 3 x ˆΨ h 2 2 } (x) 2m + V ext (x) ˆΨ(x) + g 2 ˆΨ (x) ˆΨ (x) ˆΨ(x) ˆΨ(x) Two-mode field ˆΨ(x) = χ 1 (x)â 1 + χ 2 (x)â 2 Ĥ SS = ε 11 ˆN + [ε12 + gt 1 (N 1 1)](â 1â2 + â 2â1) + gt 0 2 ( ˆN 2 1 + ˆN 2 2 ˆN) 1=L 2=R + gt 2 2 (â 1â 1â2â 2 + â 2â 2â1â 1 + 4 ˆN 1 ˆN2 )
Diagonalization of Ĥ SS Solve Ĥ SS Ψ N = E Ψ N in Fock basis N Ψ N = C N1 N 1, N 2 = N N 1 N 1 =0 Matrix equations N N 1 =0 H SS N 1 N 1C N 1 = C N1 E S-S 1 particle Schrödinger matrix elements ε kl = d 3 xχ k (x){( h 2 /2m) 2 + V ext (x)}χ l (x) T 0 = d 3 xχ 4 1(x) T 2 = d 3 xχ 2 1(x)χ 2 2(x) T 1 = d 3 xχ 3 1(x)χ 2 (x)
Parametrization of Matrix Elements (Bose-Hubbard Model) Mahmud, Perry, Reinhardt (PRA, 2005) Study energy levels as a function of barrier height α for g = 1 Mean field and single particle energies T 0 = ε LL = ε RR = 1 Lowest order tunneling ε LR = T 1 = exp( α) Higher order tunneling T 2 = exp( 2α) 0 Diagonalize Ĥ SS (α)
Energy Level Correlation Diagram ĤSS(α) doubly degenerate pairs Eigenvalues of nondegenerate delocalized states α with N = 20
Low Barrier/Energy Ψ N Coefficients C NL N L /N for N = 20
High Barrier/Energy Ψ N Coefficients Ψ 17, 3 + 3, 17 C NL N L /N for N = 20
Analogous to Physical Pendulum rotating pendulum With n = N L N R and θ is the phase difference between L,R condensates Recover Anderson s oscillator model for the Josephson effect n oscillating pendulum rotating pendulum θ
Project Ψ N into (n, θ) Phase Space n θ
Phase Imprint or Offset Spawns Solitons in BECs
Use Phase Imprint Technology in Double Well BECs Shine far detuned light in ONE well Introduce phase offset θ between wells Let wavepacket time evolve
Ground State at Zero Barrier C N1 2 n N 1 θ
Phase Imprints n θ
short time n Time Evolve to Cat-Like State C N1 2 α time n long time θ N 1 Ψ N N, 0 + 0, N
Extension to Multiple Wells Mahmud, Leung, Reinhardt (submitted to PRA) C N1,N 2 2 C N1,N 2 2 ground state t=0.22 ms C N1,N 2 2 C N1,N 2 2 t=0.37 ms t=0.68 ms N = N 1 + N 2 + N 3
PART 2 Inclusion of Mean Field Effects 1-particle orbitals computed from HF Not from Schrödinger equation No parameters
State Vector of Single Component Particle number N = N 1 + N 2 Recall multiconfiguration state Ψ N = C 0 0, N + C 1 1, N 1 + + C N N, 0 Single boson Fock state N 1, N 2 = (â 1 )N 1 (â 2 )N 2 N1 N 2 vac Symmetric product wavefunction for N 1, N 2 Ψ N 1,N 2 (1,..., N) = S{χ 1 (x 1 ) χ 1 (x N1 )χ 2 (x N1 +1) χ 2 (x N )}
Hartree-Fock Theory for Bosons Extremize the functional F [χ 1, χ 2 ] = N 1, N 2 Ĥ µ kl ˆNk ( χ 1 χ 2 δ kl ) N 1, N 2 kl=1,2 Coupled 2-mode Hartree-Fock equations hχ 1 + (N 1 1)Γ 1 χ 1 + N 2 [J 2 + K 2 ]χ 1 = µ 11 χ 1 + µ 12 χ 2 hχ 2 + (N 2 1)Γ 2 χ 2 + N 1 [J 1 + K 1 ]χ 2 = µ 21 χ 1 + µ 22 χ 2 Γ k (x)χ k (x) = d 3 x [χ k(x )V (x, x )χ k (x )]χ k (x) J k (x)χ k (x) = d 3 x [χ k (x )V (x, x )χ k (x )]χ k (x) K k (x)χ k (x) = d 3 x [χ k (x )V (x, x )χ k (x )]χ k (x)
Two-Mode Model with HF orbitals Two-mode field ˆΨ(x) = χ 1 (x)â 1 + χ 2 (x)â 2 Many-body Hamiltonian Ĥ = { d 3 x ˆΨ h 2 2 } (x) 2m + V ext (x) ˆΨ(x) + 1 d 2 3 xd 3 x ˆΨ (x) ˆΨ (x )V (x x ) ˆΨ(x ) ˆΨ(x) Most general symmetric state N Ψ N = C N1 N 1, N 2 = N N 1 N 1 =0
Diagonalization of Ĥ Solve Ĥ Ψ N = E Ψ N N N 1 =0 H N1 N 1 C N 1 in basis = C N 1 E Numerical scheme: 1. Solve HF equations for configuration N 1, N 2 2. Use HF solutions to build H N1 N 1 3. Diagonalize and repeat for new N 1, N 2 4. Use variational principle to determine optimal states
Typical HF 1-Particle Orbitals χ 1, χ 2 versus x no barrier
χ 1, χ 2 Typical HF 1-Particle Orbitals versus x high barrier
Energy versus configuration N 1 N 2. E N 1 /N for N = 20 at low barrier
Energy Level Correlation Diagram E barrier height
Low Barrier/Energy Ψ N Coefficients C N1 N 1 /N for N = 20
High Barrier/Energy Ψ N Coefficients Ψ 17, 3 + 3, 17 C N1 N 1 /N for N = 20
Contact Potential Replacement Naïvely replace χ 1 χ 2 V χ 1 χ 2 g 4π h 2 a m δ(x x ) χ 1 χ 2 V χ 2 χ 1 4π h 2 a m δ(x x ) HF equations with pseudopotential hχ 1 + g(n 1 1) χ 1 2 χ 1 + 2gN 2 χ 2 2 χ 1 = µ 11 χ 1 + µ 12 χ 2 hχ 2 + g(n 2 1) χ 2 2 χ 2 + 2gN 1 χ 1 2 χ 2 = µ 21 χ 1 + µ 22 χ 2 Note factor of 2 in interaction term
Contact Potential Replacement For two component SPINOR condensates χ 1 χ 2 V χ 1 χ 2 χ 1 χ 2 V χ 2 χ 1 4π h 2 a m δ(x x ) Esry et al, PRL 1997 hχ 1 + g(n 1 1) χ 1 2 χ 1 + gn 2 χ 2 2 χ 1 = µ 11 χ 1 hχ 2 + g(n 2 1) χ 2 2 χ 2 + gn 1 χ 1 2 χ 2 = µ 22 χ 2
Typical HF 1-Particle L,R Orbitals χ L, χ R versus x no barrier
Typical HF 1-Particle L,R Orbitals χ L, χ R versus x high barrier
Energy versus configuration N 1 N 2. E N 1 /N for N = 20 at low barrier
Energy Level Correlation Diagram E barrier height
Typical HF 1-Particle S,A Orbitals no barrier χ S, χ A versus x high barrier
Energy versus configuration N 1 N 2. E N 1 /N for N = 20 at low barrier
Energy Level Correlation Diagram E barrier height
Fragmented Ground State for Large Coupling Constant Consequence of factor of 2 χ 1, χ 2 x
Thanks to Bill Reinhardt, UW Chemistry and Physics Khan Mahmud, UM Physics Heidi Perry, Columbia Chemistry Mary Ann Leung, UW Chemistry Sam McKagan, JILA and CU Boulder Physics NSF-Physics
EXTRA SLIDES
Linear Combination CN1 ± C N1+1 N 1 /N for N = 20