Laguerre-Gaussian laser modes for atomic physics Atom channeling - PowerPoint PPT Presentation

Laguerre-Gaussian laser modes for atomic physics Atom channeling and information storage QTC 2015 Laurence PRUVOST Laboratoire Aimé Cotton, CNRS, Univ. Paris-Sud, ENS Cachan Orsay, France Ph 740-12

Twisted light: Atom channeling and information storage Introduction : twisted light and orbital angular momentum Laguerre-Gaussian modes properties. preparation, methods to detect Use of the ring shape cold atoms: LG-Channelled-2D MOT Use of the quantum number of the phase generalities OAM storage/retrieval in cold atoms by Four Wave Mixing and Coherent Population Oscillation | m F > ω Ω ’ Ω Ω C Ω R Conclusion | m F -1> | m F +1> C R z' W’ ’

Twisted light /vortex beam Def : EM wave imprinted/carrying an helical phase LG04 Properties dark center for ℓ≠0  Fresnel diffraction and symmetry Orbital angular momentum (OAM), L= ћℓ  Helical phase ℓ quantum number of the beam  Classed in families of solutions of the Helmholtz (paraxial) equation ex. Laguerre-Gaussian modes 3

Orbital Angular Momentum  L. Allen et al., Phys. Rev. A 45, 8185 (1992).  The associated OAM is  L characterizes how the phase turns  OAM quantized with the signed integer ℓ , varying from –∞ to +∞. ℓ also called mode order, or azimuthal number . OAM differs from the linear momentum k and the polarization (SAM) 4

Laguerre-Gaussian modes 5

Laguerre-Gauss Modes Eigen solutions of the paraxial wave equation Hermite-Gauss TEM mn Laguerre-Gauss LG ℓ p Cartesian coordinates cylindrical coordinates

Laguerre-Gauss Modes Eigen solutions of the paraxial wave equation  Propagates keeping the shape  LG modes constitute a basis  a LG  , light p p  , p  LG ℓ p ℓ : OAM, azimuthal number p : radial number

E= Amplitude factor Helical phase wavefront curvature Modulated by Laguerre Polynomial Gouy phase Gaussian envelope  Inside the Gaussian envelope the intensity has p+1 rings LG modes. Details, properties p=0 p=1 p=2 p=3  The center of the p=0 varies ρ 2 ℓ power-law ℓ =1 harmonic ℓ >>1 squared 8

E= Amplitude factor Helical phase wavefront curvature Gaussian envelope Gouy phase  phase exp[ i ℓ θ ] . The electric field is angular dependent and changes of sign ℓ times (figure for ℓ=4, p=0,1,2,3) LG modes. PHASE properties Rings of helical phases 9

LG preparation and detection 10

LG generation  Imprint a helical phase to an input beam Use of a spatial light modulator ( SLM ) as phase mask 2π 0 11

Spatial Light Modulator. Phase-only SLM  SLM = Programmable 2D optical component able to modify the amplitude and/or the phase of the light at each point of its surface. • Micro mirror devices, deformed mirrors, liquid crystal devices  In liquid-crystal SLM, the LC molecules are oriented by an electric field map, changing locally the birefringence (the index) and thus the phase of a beam going through. V n e n o SLM Hamamatsu, active area : 2cm x2 cm 12

Our setup Detected on the CCD ℓ =10 Gaussian beam SLM ℓ =100 Rem : very dark center

 LG ℓ=10 p=0 ?  Very dark center, close to ρ 2 ℓ shape ℓ=10  Thin principal ring  extra rings due p≠0 components

Other methods of fabrication  dark spot (absorbent) in a laser cavity  conversion of HG to LG using a set of cylindrical lenses  amplitude mask, being a fork pattern picture : mask used for electron vortex beams .  wavefront imprinting by a vortex phase, by a shaped plate by an holographic plate ℓ is easily changed by SLM

Methods for OAM detection Ring shape not enough. Phase detection needs interferences 1. Phase analyser Phase analyzer, Shark-Hartmann or micro lenses 2. Interference with a reference beam T wisted pattern to analyzed 3. Diffraction by an aperture (double slit, triangle, wheel- hole… ) ℓ=0 ℓ=1 ℓ=2 L Pruvost: LG modes in atomic physics 2015 16

OAM detection 4. Self-interferences with a astigmatic sytem  As the lens is turned : Self-interference with ℓ dark lines  OAM determined by the fringe number : Cf : Vaity et al. Phys. Lett. A 377: 1154-1156, 2013

LG modes use Ring shape OAM (phase)  STED microscopy  Transfer to objects i.e. rotation cf Grier, cf Hell, Moermer  Transfer to atomic waves  particule traps=optical tweezers cf Philips cf Grier,  Optics / Non-linear optics  dipole potential (squared) cf Padgett, Zeilinger cf Hazebebics  Information encoding cf Wang, Tamburini, Bo Thidé > 0 U ℓ =10, 7, 13 Atoms in the LG dark center superposition 18

Use of ring shape LG-2D-MOT V Carrat, C Cabrera, M Jacquey, J R W Tabosa, B Viaris de Lesegno, LP Experiment done in Orsay (France ) cf C. Cabrera talk, Thursday 10:40 19

“Light tube” to channel the atoms  Atoms exiting a 2D-MOT used to load a 3D-MOT  Channel the atoms, reduces the divergence and increases the density reduce the capture zone and efficient loading 20

| m F > ω Ω Ω ’ Ω C Ω R | m F -1> | m F +1> C R z' W’ ’ Use of the helical phase Storage/retrieval of OAMs in cold atoms R.A. de Oliveira, L. Pruvost, P.S. Barbosa, W.S. Martins, S. Barreiro, D. Felinto, D. Bloch, J.W.Tabosa Experiments done in Recife (Brazil) Applied Physics B 2014, Optics Letters 2015. 21

Context  Orbital angular momentum of light as a variable for encoding  Using an atomic system – as simple as possible - many groups explore the writing, storing, reading OAM.  Atoms are a simple model to experience and explain the involved ω Ω ’ Ω Ω Ω processes. Cold atoms Laser encoding laser z OAM Analysis of the output W’ 22 ’ See Also : E Giacobino & J Laurat, Paris, France, Nat, Phot, 8, 234, 2014 G-C Guo, Hefei 230026, China, Nat, Com, 4, 252, 2013 S Franke-Arnold, Glasgow, UK,

Principle of the OAM storage/retrieval  The information is encoded on W (or W’) beam (Laguerre -Gaussian mode)  The memory is a cold atom sample. Writing information Reading information  The information is retrieved via four wave mixing (FWM) process.

Interaction with the atom : Λ system. Dark state e  Atom submitted to 2 lasers W, W’ each realizing a transition. The Rabi frequencies are Ω and Ω’. W W’  Interaction matrix in the initial natural basis g 1 , g 2 , e is V (g1,g2,e) g 1 g 2  In a new basis (dark, bright, e) ' 0 0 1 2 dark g g 2 2 2 2 ' ' 0 0 ' V ( 1 , 2 , ) g g e ' ' 0 1 2 bright g g 2 2 2 2 ' ' 0 0 0 the dark state (DS) is not coupled to e state. Any atom falling into 0 0 V eff ( , , ) dark bright e the DS, remains in it and 0 0 eff becomes insensitive to the light

Phase sensitive interaction of the dark state e  The DS combines 2 ground states, having a long lifetime . In principle, it lives for a long W W’ time.  the DS combines the Rabi g 1 g 2 frequencies Ω, Ω’. So it contains information carried by Ω and Ω ’ . ' 1 2 dark g g It contains the relative phase  2 2 2 2 ' ' of W and W’, so the OAM . . W ' . ' W

OAM writing, reading Another point of view : W, ℓ  The interference of W and W’ making an angle θ, creates a fork coherence pattern , W’ , imprinted in the cold atom sample. Coherence phase pattern 2   2 sin( / 2 ) k k F P  The reading beam R diffracts on the fork pattern.  The emitted beam (C) acquires the OAM and propagates with an angle θ with R. R, C, ℓ

Storage and non-collinear retrieval of OAM (A)$ | m F > | m F > 6P 3/2 , F=2 OAM Measurement of the ω 0 Ω w Ω w ’ Ω C Ω R emitted beam 6S 1/2 , F=3 | m F -1> | m F +1> | m F -1> | m F +1> C W W z R z' W’ W’ W, W ’ ON OFF t s R ON OFF Time 0 The diffracted beam C is emitted in a direction different from the initial input direction. Conservation of the OAM is observed without or with time delay. The angle θ is small (2 ) & ℓ is small.

| m F > ω Ω Ω ’ Ω C Ω R | m F -1> | m F +1> C R z' W’ Delayed FWM realizes OAM ’ storage/retrieval in cold atoms Storage of angular momentum of light OAM and  collinear and Off-axis retrieval. Revieval using Larmor oscillations  Off-axis retrieval of orbital angular momentum of light stored in cold atoms, R A de Oliviera L Pruvost, PS Barbosa, WS Martins, S Barreiro, D Felinto, D Bloch, J WR Tabosa. Appl. Phys. B 17 , 1123-1128 (2014) 28

OAM storage/ retrieval with CPO Polarisation configurations EIT : σ + / σ - e F’=0 F=1 M=+1 M=-1 CPO : lin  lin (Coherent Population Oscillations) e lin = σ + + σ - F’=0  lin = σ + - σ - Eq. to 2 two-levels systems F=1 +1 -1 29

OAM storage/ retrieval with CPO Polarisation configurations EIT : σ + - σ - e F’=0 F=1 M=+1 M=-1 CPO : lin  lin e F’=0 F=1 CPO efficient and robust against +1 magnetic field -1 30

EIT/ CPO memories  CPO robust against magnetic field  CPO storage longer than EIT ones  Also observed in hot vapour Cs:Tabosa, 2014 He: Laupretre, Goldfarb 2014 L Pruvost: LG modes in atomic physics QTC-2015 31

OAM storage / retrieval with CPO  One beam carrying OAM  Collinear and off-axis case L Pruvost: LG modes in atomic physics QTC-2015 32