Laboratoire Kastler Brossel Collège de France, ENS, UPMC, CNRS Introduction to Ultracold Atoms Optical lattices – Atoms in artificial crystals made of light Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr ) Advanced School on Quantum Science and Quantum Technologies, ICTP Trieste September 5, 2017
From one to three-dimensional optical lattices A coherent superposition of waves with different wavevectors results in interferences. The resulting interference pattern can be used to trap atoms in a periodic structure. Linear stack of 2D gases Planar array of 1D gases Cubic array Additional, weaker trapping potentials provide overall confinement of the atomic gas. Many more geometries are possible by “playing” with the interference patterns. Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Goal of the next three lectures Why is this interesting ? 1 Connection with solid-state physics (band structure and related phenomenon) 2 A tool for atom optics and atom interferometry: coherent manipulation of external degrees of freedom 3 Path to realize strongly correlated gases and new quantum phases of matter In the next lectures, we will discuss the behavior of quantum gases (mostly bosons, a little about fermions) trapped in periodic potentials. Outline 1 A glimpse about experimental realizations, and single-particle physics : band structure, Bloch oscillations. 2 Superfluid-Mott insulator transition for bosonic gases Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
1 Realizing optical lattices 2 Band structure in one dimension 3 BECs in optical lattices 4 Bloch oscillations Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
One-dimensional optical lattices Superposition of mutually coherent plane waves : 2 � � | E ( r ) | 2 = � � | E n | 2 + � � � E n e i k n · r E ∗ n · E n ′ e i ( k n ′ − k n ) · r = � � � � � n � n n � = n ′ Intensity (and dipole potential) modulations with wavevectors k n ′ − k n Simplest example : • Standing wave with period d = π/k L k L = λ L π d = 2 + k L − k L • Trapping potential of the form (red detuning): V ( x ) = − 2 V 1 (1 + cos(2 k L x )) position x = − V 0 cos( k L x ) 2 For red detuning (the case assumed by default from now on for concreteness), atoms are trapped near the antinodes where V ≈ − 4 V 0 . Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Two and three-dimensional optical lattices Two dimensions : | E ( r ) | 2 ≈ | 2 E 0 cos( k L x ) | 2 + | 2 E 0 cos( k L y ) | 2 (a) (b) (c) − k L e z , ω 3 + k L e y , ω 2 + k L e y , ω 2 − k L e x , ω 1 + k L e x , ω 1 − k L e x , ω 1 − k L e y , ω 2 + k L e x , ω 1 − k L e y , ω 2 y x z + k L e z , ω 3 y x Two-dimensional square potential Three-dimensional cubic potential ν =1 , ··· ,d − V ν cos( k ν x ν ) 2 Square ( d = 2 ) or cubic ( d = 3 ) lattices : V lat ( r ) = � • Separable potentials : sufficient to analyze the 1D case Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Triangular/honeycomb optical lattices Other lattice geometries are realizable as well. Example with three mutually coherent coplanar beams [ Soltan-Panahi et al. , Nature Phys. (2011) ]: k 3 e x , ω 1 k 1 e x , ω 1 k 2 e x , ω 1 y x • intensity maxima on a triangular lattice • intensity minima on a honeycomb lattice (triangular with two atoms per unit cell) More complex example : the Kagomé lattice [ Jo et al. , PRL (2012) ]: Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
One-dimensional lattice k L = λ L π d = • Standing wave with period d = π/k L 2 + k L − k L • Trapping potential : V ( x ) = V 0 sin 2 ( k L x ) position x 87 Rb, λ L = 1064 nm: Natural units : • lattice spacing d = λ L / 2 = π/k L • d ≈ 532 nm • recoil momentum � k L • E R ≈ h × 2 kHz ( k B × 100 nK) • recoil energy E R = � 2 k 2 L 2 M Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Diffraction from a pulsed lattice • Apply a lattice potential on a cloud of ultracold atoms (BEC) for a short time, • look at momentum distribution : Lattice axis 2 k L 0 µ s 4 µ s 8 µ s 12 µ s 16 µ s 20 µ s 24 µ s 28 µ s Pulse duration • initial BEC : narrow wavepacket in momentum space (width ≪ k L ) • treat it as plane wave with momentum k = 0 Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Diffraction from a pulsed lattice : Kapitza-Dirac regime Pulsing a lattice potential on a cloud of ultracold atoms (BEC) : Lattice axis 2 k L 0 µ s 4 µ s 8 µ s 12 µ s 16 µ s 20 µ s 24 µ s 28 µ s Pulse duration Raman-Nath approximation: • BEC → plane wave with k = 0 1.0 p=0 p=2 • neglect atomic motion in the potential p=1 Raman-Nath 0.8 Relative populations during the diffraction pulse : 0.6 Ψ( x, t ) ≈ e i V 0 cos(2 kLx ) t 0.4 Ψ( x, 0) 2 � + ∞ 0.2 � J p ( V 0 t/ 2 � ) e i 2 pk L x . ≈ 0.0 p = −∞ 0 10 20 30 40 50 Pulse duration ( µ s) J p : Bessel function Raman-Nath approximation valid only for • Analogous to phase modulation of short times light wave by a thin phase grating Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
1 Realizing optical lattices 2 Band structure in one dimension 3 BECs in optical lattices 4 Bloch oscillations Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Bloch theorem Hamiltonian : Natural units: • lattice spacing d = λ L / 2 = π/k L p 2 ˆ ˆ H = 2 M + V lat (ˆ x ) • recoil momentum � k L V lat ( x ) = − V 0 sin 2 ( k L x ) • recoil energy E R = � k 2 L / 2 M Lattice translation operator : • definition : ˆ T d = exp ( i ˆ pd/ � ) • � x | ˆ T d | φ � = φ ( x + d ) for any | φ � • [ ˆ T d , ˆ H ] = 0 . Bloch theorem : Simultaneous eigenstates of ˆ H and ˆ T d ( Bloch waves ) are of the form φ n,q ( x ) = e iqx u n,q ( x ) , where the u n,q ’s ( Bloch functions ) are periodic in space with period d . • q : quasi-momentum • n : band index ˆ Hφ n,q ( x ) = ε n ( q ) φ n,q ( x ) , T d φ n,q ( x ) = e iqd φ n,q ( x ) , ˆ Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Bloch theorem Bloch waves : φ n,q ( x ) = e iqx u n,q ( x ) , where the u n,q ’s ( Bloch functions ) are periodic in space with period d . • q : quasi-momentum • n : band index • Periodic boundary conditions for a system with N s sites (length L = N s d ) : q j = 2 π j L j = 2 k L N s with j ∈ Z , | j | ≤ N s / 2 State labeling : • Quasi-momentum is defined from the eigenvalue of ˆ T d : T d φ n,q ( x ) = e iqd φ n,q ( x ) . ˆ • For Q p = 2 pk L with p integer (a vector of the reciprocal lattice ), ˆ T d φ n,q + Q p ( x ) = e i ( q + Q p ) d φ n,q + Q p ( x ) = e iqd φ n,q + Q p ( x ) . • To avoid double-counting, restrict q to the first Brillouin zone : BZ1 = [ − k L , k L ] . Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Fourier decomposition of Bloch waves on plane waves Bloch waves : φ n,q ( x ) = e iqx u n,q ( x ) The Bloch function u n,q is periodic with period d : Fourier expansion with harmonics Q m = 2 mk L of 2 π/d = 2 k L . � u n,q ( m ) e iQ m x , u n,q ( x ) = ˜ m ∈ ❩ V lat ( m ) e iQ m x = − V 0 2 + V 0 � e iQ − 1 x + e iQ 1 x � � ˜ V lat ( x ) = 4 m ∈ ❩ • the Bloch functions are superpositions of all harmonics of the fundamental momentum 2 k L . • the lattice potential couples momenta p and p ± 2 k L . Useful to solve Schrödinger equation : reduction to band-diagonal matrix equation for the Fourier coefficients ˜ u n,q ( m ) (tridiagonal for sinusoidal potential) Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Spectrum and a few Bloch states, V 0 = 0 E R Free particle spectrum : ǫ n ( q ) = � 2 ( q +2 nk L ) 2 , Momentum : k = q + 2 nk L 2 M Degeneracy at the edges of the Brillouin zone : E n ( ± k L ) = E n +1 ( ± k L ) Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Spectrum and a few Bloch states, V 0 = 2 E R Gaps open near the edges of the Brillouin zones ( q ≈ ± k L ) Lifting of free particle degeneracy by the periodic potential Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Spectrum and a few Bloch states, V 0 = 4 E R Gaps widen with increasing lattice depth V 0 Bands flatten with increasing lattice depth V 0 Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Spectrum and a few Bloch states, V 0 = 10 E R Gaps widen with increasing lattice depth V 0 Bands flatten with increasing lattice depth V 0 Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
Diffraction from a pulsed lattice from band theory Pulsing a lattice potential on a cloud of ultracold atoms (BEC) : Lattice axis 2 k L 0 µ s 4 µ s 8 µ s 12 µ s 16 µ s 20 µ s 24 µ s 28 µ s Pulse duration Bloch wave treatment: ∞ p=0 Bloch waves 1.0 � | φ n,q � = u n,q ( m ) | q + 2 mk L � ˜ p=1 Raman-Nath p=2 Relative populations 0.8 m = −∞ � k = 0 | φ n,q � = ˜ u n,q =0 ( m = 0) 0.6 0.4 Initial state : 0.2 u n,q =0 ( m = 0)] ∗ | φ n,q =0 � � | Ψ( t = 0) � = | k = 0 � = [˜ 0.0 n 0 10 20 30 40 50 Evolution in lattice potential : Pulse duration ( µ s) Raman-Nath approximation valid only for En,q =0 t short times u n,q =0 ( m = 0)] ∗ e − i � | Ψ( t ) � = [˜ | φ n,q =0 � � n Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
1 Realizing optical lattices 2 Band structure in one dimension 3 BECs in optical lattices 4 Bloch oscillations Fabrice Gerbier ( fabrice.gerbier@lkb.ens.fr )
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