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8. Superfluid to Mott-insulator transition Overview Optical lattice potentials Solution of the Schrödinger equation for periodic potentials Band structure Bloch oscillation of bosonic and fermionic atoms in optical lattices Wannier functions Bose-Hubbard Hamiltonian Superfluid to Mott-insulator transition Optical dipole traps ϕ 1 1 Γ Potential: U Dip (r ) = d ⋅ E t = Re(α ) I ( r ) ∝ I (r ) 2 2ε 0c Δ 2 P 1 ⎛Γ⎞ Loss rate: γ sc (r ) = abs = Im(α ) I ( r ) ∝ ⎜ ⎟ I (r ) =ω =ε 0 c ⎝Δ⎠ d: induced dipole moment, α: polarizability abs ω0 ω Γ: Dipole matrix element between ground and excited states, Δ=ω−ω0: detuning Δ <0 : red detuned Δ >0 : blue detuned attractive potential repulsive potential Gaussian beam: I (r ) = I 0 e − 2r2 w2 Optical lattices Optical standing wave fields produce periodic potentials with lattice constants half of the laser’s wavelenght (λ ~ 800 nm – 10 μm). 1-D 2-D ⎛ 2π ⎞ I ( x ) = I 0 sin 2 ⎜⎜ x + ϕ ⎟⎟ ⎝ λlaser ⎠ a= λlaser 2 3-D Band model Solutions of the Schrödinger equation in periodic potentials SE: Ansatz: Comparing coefficients yields conditional equations for parameters C(k) For a specific k ∈ [-k0/2, k0/2] the wavefunction ψk contains also wavevectors k+nk0 . In order to find energies and eigenstates for a given k, the equation system (*) has to be solved for all k ∈ [k+nk0] . Note: |k> is not an eigenstate of the Hamiltonian. Band model – energies and eigenstates system Band model – example Dynamics in a lattice potential Phase difference between neighboring lattice sites Bloch-Oscillation (1D lattice) ϕ j − ϕi = (E j − Ei ) ⋅ t = Nonlinear dynamics leads to dephasing if gradient is left on for longer times ! (2D lattice) Δφ = 0 Δφ = π M. Greiner et al. PRL 87, 160405 (2001) Bloch oscillation of fermionic atoms Large contrast for mor than 100 oscillation periodes Comparison: fermions vs. bosons (a) Momentum distribution of fermions in the lattice: 1 ms (continuous line) and 252 ms (dashed line). (b) Momentum distribution of bosons at 0.6 ms (continuous line) and 3.8 ms (dashed line). The much faster broadening for bosons is due to the presence of interactions. TB=1.2 s G. Roati et al., Phys. Rev. Lett. 92, 230402 (2004) Fermi surfaces Bose gas in a 3D lattice potential Resulting potential consists of a simple cubic lattice where the BEC coherently populates about 100,000 lattice sites. M. Greiner et al. PRL 87, 160405 (2001) Interference of a superfluide bose gas from a 3D lattice M. Greiner et al. PRL 87, 160405 (2001) Wave function of single particles in a periodic potential Bloch states: Ψ k (r) = u k (r)eikr Plane waves modulated by a lattice periodic function Fourier transform 1 BZ ikR j Wannier states: w(r − R j ) = e Ψ k (r) ∑ L k Localized wave functions. This pictures allows tunneling as well as localized states. Inverse Fourier transform 1 lattice sites − ikR j e w(r − R j ) Ψ k (r) = ∑ L j The Hamiltonian can be written in Wannier basis (TONS). Second quantization results in the Bose-Hubbard Hamiltonian. (Necessary condition: only the lowest band is populated ↔ excitation energies are larger than the energy gap.) Bose-Hubbard Hamiltonian Expanding the field operator in the Wannier basis of localized wave functions on each lattice site, yields : Bosonic operators ψˆ ( x ) = ∑ aˆi w( x − xi ) i annihilation â i ..., N i ,... = N i ..., N i − 1,... creation â i+ ..., N i ,... = N i + 1 ..., N i + 1,... ( aˆ i ) * Bose-Hubbard Hamiltonian number = aˆ i+ ⎡⎣aˆ i ,aˆ +j ⎤⎦ = δij aˆ +j aˆ j = n j 1 H = − J ∑ aˆi† aˆ j + ∑ ε i nˆi + U ∑ nˆi (nˆi − 1) 2 i i, j i Single particle energy in the trapping potential Tunnel matrix element (hopping element) Onsite interaction matrix element ⎛ =2 2 ⎞ J = − ∫ d x w( x − xi ) ⎜ − ∇ + Vlat ( x )⎟ w( x − x j ) ⎝ 2m ⎠ 4π = 2 a 3 4 U= d x w ( x ) m ∫ 3 M.P.A. Fisher et al, PRB 40, 546 (1989); D. Jaksch et al., PRL 81, 3108 (1998) Superfluid Limit The kinetic energy dominates (weakly interacting bosonic system). 1 H = − J ∑ aˆi† aˆ j + U ∑ nˆi (nˆi − 1) 2 i i, j Atoms are delocalized over the entire lattice, macroscopic wave function describes this state. Ψ SF ⎛ M †⎞ ∝ ⎜ ∑ aˆi ⎟ ⎝ i =1 ⎠ N 0 ai ≠ 0 Poissonian atom number distribution per lattice site n=1 n=2 Mott-insulator Interaction energy dominates (strongly correlated bosonic system). 1 H = − J ∑ aˆi† aˆ j + U ∑ nˆi (nˆi − 1) 2 i i, j Atoms are completely localized to lattice sites. M ( ) Ψ Mott ∝ ∏ a i † i =1 n 0 ai = 0 Fock states with a vanishing atom number fluctuation are formed, e.g. n=1. n=1 Momentum distribution for different potential depths (Erecoil) 0 Erecoil 22 Erecoil M. Greiner et al., Nature 415, 39 (2002) Restoring coherence a) Ramp up the potential for generating the Mott-insuator state and subsequent ramp down. Measurement on a phase incoherent cloud. b) Width of the zero momentum central peak (Dephasing was applied before reaching the Mott-insulator state) The coherence is restored within the tunneling time to the neigboring lattice site. Measurement on a phase incoherent cloud. Before ramping down the potential M. Greiner et al., Nature 415, 39 (2002) Quantum phase transition from a superfluid to a Mott-insulator At the critical point gc the system will undergo a phase transition from a superfluid to an insulator. This phase transition occurs even at T=0 and is driven by quantum fluctuations ! Characteristic of the quantum phase transition • Excitation spectrum is dramatically modified at the critical point. • U/J < gc (Superfluid regime) Excitation spectrum is gapless • U/J > gc (Mott-Insulator regime) Excitation spectrum is gapped Critical ratio: U/J = z * 5.8 number of next neighbors (for a cubic lattice 6) Creating excitations in the MI phase Mott-insulator with ni=1 atom per lattice site Without gradient potential With gradient potential Special case: ΔEij = U Energy Scales: =ωv ≈ 20⋅U U ≈10 − 300⋅ J Measuring the excitation gap in the MI phase (probability vs. gradient) 10 Erecoil tperturb = 2 ms 13 Erecoil tperturb = 4 ms 16 Erecoil tperturb = 9 ms 20 Erecoil tperturb = 20 ms M. Greiner et al., Nature 415, 39 (2002) Literature Bose-Einstein condensates in 1D- and 2D optical lattices M. Greiner et al. Appl. Phys. B. 73, 769 (2001) Exporing the phase coherence in a 2D lattice of Bose-Einstein condensates M. Greiner et al. PRL 87, 160405 (2001) Fermionic atoms in a three dimensional optical lattice: observing Fermi surfaces, dynamics and interactions, M. Köhl et al., PRL94, 080403 (2005) Quantum phase transition from a superfluid to a Mott insulator in a gas of ultracold atoms M. Greiner et al., Nature 415, 39 (2002)