Cold atoms in optical lattices

Transcription

1 Cold atoms in optical lattices Tarruel, Nature Esslinger group

2 Optical lattices the big picture We have a textbook model, which is basically exact, describing how a large collection of atoms will behave in free space H = H = T 2 pi 2m i + V + g ij + U i, j δ ( r r ) pair-wise contact interactions i j We can add a potential term (V), modify the kinetic energy term (T), and even get qualitatively new, effective interaction terms (U) by putting the atoms into optical lattices

3 Optical lattices the big picture One obvious analogy: motion of atomic matter waves in laser crystals ~ motion of electron waves in ionic crystal lattices Electron matter in solid-state crystals Atomic matter in light crystals ~ nm Extremely dilute less dense than H 2 O Ultracold nk and pk temperatures No disorder (no phonons or defects) λ/2 ~ μm exact microscopic model low energy scales EE 1/mmLL 2

4 Optical lattices the big picture Not just limited to studying electronic systems many problems can be mapped from continuum to lattice (example: lattice QCD)

5 Optical lattices the big picture Aside from quantum simulation / studying many-body physics, also just a great way to keep atoms from moving Campbell et al., Science 358, (2017) rid of Doppler shifts for atomic clocks

6 Optical lattices the big picture Aside from quantum simulation / studying many-body physics, also just a great way to keep atoms from moving Wang et al., Science 352, (2016) trapping of qubits

7 Optical lattices laser interference Immanuel Bloch Nature (2008) VV rr, tt = αα ωω II rr, tt αα ωω 1 ωω ωω 0 II rr, tt typically formed by laser interference

8 Optical lattices laser interference Greiner/Vuletić/Lukin collab, CUA

9 Optical lattices laser interference Saffman group, Wisconsin

10 Optical lattices laser interference Browaeys group, Palaiseau Note: such potentials have large inter-well spacing, not particularly well-suited to studying coherent tunneling (exceptions: Jochim / Regal groups)

11 Optical lattices laser interference Immanuel Bloch Nature (2008) VV rr, tt = αα ωω II rr, tt αα ωω 1 ωω ωω 0 II rr, tt = nn EE nn rr, tt εε 2 nn typically formed by laser interference Lattice pattern depends on: frequencies polarizations directions of propagation relative phases

12 Simplified 1D optical lattice Immanuel Bloch Nature (2008) VV rr, tt EE 1 ee ii kkkk ωωωω+φφ 1 + EE 2 ee ii kkkk ωωωω+φφ 2 2 VV rr, tt = VV offset + VV 0 cos 2 ππ zz zz 0 dd Note: for real laser beams (~Gaussian, not plane-wave), also get confinement (or deconfinement) in radial direction [i.e. V 0 is a slowly-varying function of rr)

13 Simplified 1D optical lattice Immanuel Bloch Nature (2008) VV rr, tt EE 1 ee ii kkkk ωωωω+φφ 1 + EE 2 ee ii kkkk ωωωω+φφ 2 2 VV rr, tt = VV offset + VV 0 cos 2 ππ zz zz 0 dd = VV offset + VV 0 2 cos 2ππzzz dd Note: for real laser beams (~Gaussian, not plane-wave), also get confinement (or deconfinement) in radial direction [i.e. V 0 is a slowly-varying function of rr)

14 Relevant energy scale The recoil energy --- for counter-propagating beams, EE RR = ħ2 kk LL 2 2mm 2θθ EE RR = h2 2mmdd 2 ss = VV 0 /EE RR dd = λλ 2 sin θθ

15 The energy band structure ss = VV 0 EE RR = 5 ss = 5 Blue: extended zone scheme energy bands vs. momentum w.r.t. crystal Black: folded zone scheme band structure

16 The energy band structure ss = VV 0 EE RR = 5 ss = 5 Blue: extended zone scheme energy bands vs. momentum w.r.t. crystal Black: folded zone scheme band structure Bloch wave functions Φ nn qq zz n = band index q = quasimomentum Note: for sinusoidal potentials, there is an exact, analytical solution for the Bloch wavefunctions / energies in terms of the Mathieu equation (characteristic Mathieu functions)

17 Localized Wannier orbitals ss = 5 ww nn zz zz jj = 1 NN qq ee iiiiii/ħ Φ qq nn zz

18 Deep lattice (s >> 1), harmonic approx. ss 2 EE RR cos 2kkkk CC + ssee RR 4 2kkkk 2 CC mmωω2 zz 2 ħωω 2 ssee RR Can approximate the Wannier states as HO orbitals (Gaussian wfs) with πσσ/dd ss 1/4 This approximation is valid, for some HO level n, for E n << se R

19 Moving optical lattices ωω 1 ωω 2 Immanuel Bloch Nature (2008) VV rr, tt ssee RR cos 2 ππzz dd + φφ 2 φφ 1 2 A sudden phase jump by ππ/2 between the two fields will shift the lattice by ¼ of a wavelength A continuous linear phase shift will make the lattice move with some fixed velocity --- this is equivalent to a fixed frequency detuning between the interfering beams

20 Moving optical lattices ωω 1 ωω 2 Immanuel Bloch Nature (2008) VV rr, tt ssee RR cos 2 ππzz dd + φφ 2 φφ 1 2 A sudden phase jump by ππ/2 between the two fields will shift the lattice by ¼ of a wavelength A continuous linear phase shift will make the lattice move with some fixed velocity --- this is equivalent to a fixed frequency detuning between the interfering beams vv llllllll = ff λλ/2

21 The energy band structure ss = VV 0 EE RR = 5 What would happen to atoms if you suddenly turned on a moving lattices, such that v latt = v R, or k = k L?

22 Diffraction from moving lattices Incoming Laser Field 87 Rb BEC Retro Laser Field EE/ħ ee large ωω + ωω 0,1 Laser spectrum ωω + ωω 0,1 gg p/2ħkk ωω + ωω 0,1 δδδδ

23 Diffraction from moving lattices Incoming Laser Field 87 Rb BEC Retro Laser Field EE/ħ ee large ωω + ωω 0,1 Laser spectrum ωω + ωω 0,1 gg p/2ħkk ωω + ωω 0,1 δδδδ Two-photon Bragg transition

24 What about applying a force? E q ( x) = V Fx V latt + φφ tt tt 2

25 What about applying a force? E Nägerl group, Innsbruck q ( x) = V Fx V latt + φφ tt tt 2

26 Release of atoms from a lattice One typically sees a diffraction pattern the weights of the different momentum orders relates to the projection of the Bloch states onto free particles states

27 Band-mapping Map population in different lattice bands to unique momentum states. Slow (adiabatic) with respect to band gap, but fast with respect to bandwidth Greiner, et al.

28 Higher-dimensional lattices Immanuel Bloch Nature (2008) Immanuel Bloch Nature (2008)

29 Higher-dimensional lattices (lower-dimensional systems ) Bloch, Dalibard, Zwerger (2008)

30 Optical superlattices Bloch group, Porto group, others

31 Multi-beam lattice structures

32 Multi-beam lattice structures Tunable 4-beam lattice (Porto group)

33 Multi-beam lattice structures Tunable 6-beam lattice (Esslinger group)

34 Triangular lattices Sengstock group

35 Triangular lattices Sengstock group

36 Kagome lattices Stamper-Kurn group