Quantum information processing with individual neutral atoms in optical tweezers. Philippe Grangier. Institut d Optique, Palaiseau, France

Transcription

1 Quantum information processing with individual neutral atoms in optical tweezers Philippe Grangier Institut d Optique, Palaiseau, France

2 Institut d Optique Théorique et Appliquée, Orsay Institut d Optique Graduate School, Palaiseau Paris Orsay Palaiseau May 20 th 2007 May 30 th 2007 Sept. 30 th 2007

3 The single atom(s) project(s) Jérôme Beugnon Benoît Darquié Matt Jones Jos Dingjan Harold Marion Gaetan Messin Andrew Lance Yvan Sortais Philippe Grangier Antoine Browaeys Tatjana Wilk Charles Tuchendler A. Fuhrmanek Charles Evellin Alpha Gaétan Y. Miroschnychenko

4 Quantum Information Processing with Individual Neutral Atoms in Optical Tweezers. Lecture 1 : trapping and moving single atoms Lecture 2 : using single atoms as controlled single-photon sources Lecture 3 : encoding qubits on single atoms, and entangling them using Rydberg blockade.

5 Lecture 1 : trapping and moving single atoms 1. Basics of neutral atoms cooling and trapping 2. Manipulating single atoms in optical tweezers

6 Laser manipulation of atoms A. Radiative forces Momentum exchanges between atoms and photons Atomic polarisability Two types of force B. Resonant radiation pressure Properties Stopping atoms with light C. Dipole force Properties Optical tweezers and lattices D. Cooling: optical molasses Doppler cooling Limit temperature Below the Doppler limit 1

7 Laser manipulation of atoms. A remarkable illustration of laser-atom interaction From the early 1980 s, from basic research (Nobel Prize 1997 for laser cooling) to applications (atom interferometers sensitive to inertial and gravitational effects; ultra-cold atom clocks) A case study showing the interest of diversified theoretical treatments: Semi-classical approach: only the atom is quantized Fully quantum approach: light also is quantized Lead to the emergence, in 1995, of a new research field : Bose- Einstein condensation of atomic gases (Nobel Prize 2001) including atom lasers: concepts similar to the ones developed in the case of photon lasers!

8 Radiation pressure and linear momentum exchange between photons and atoms Linear momentum balance sheet in a fluorescence cycle Photon in a running wave with k-vector : p = ħk E b Absorption-Reemission ΔP at = ħk ħk sp Average over many cycles Average force : k sp = 0 ΔP at = ħk 1 cycle E a F res =N ΔP at 1 cycle = N ħk = ħk Γ 2 s 1 + s s = Ω 2 1 /2 (ω 0 ω) 2 +Γ 2 /4 Orders of magnitude Rb Γ /2π = 6 MHz I sat = 1.6 mw / cm 2 max F max M = ħk M Γ m / s g 3

9 Application: stopping an atomic beam Sodium atomic beam V 2k B T M 600 m / s with a laser stopping distance = V 2 2γ = 1.8 m Atoms must be kept in resonance during slowing (Doppler effect compensation): position depending Zeeman shifts Zeeman slower (Phillips and coll., 1982) Velocity distribution Density in velocity space increases: genuine cooling 4

10 Zeeman slowing of metastable Hélium 5

11 Radiative forces : general approach Forced oscillation of the atomic dipole, under the effect of the field ˆD ε E(r,t) = ε E 0 (r)cos ωt ϕ(r) with E(r,t) = = ε 0 α E(r,t) + ε 0 α * E * (r,t) ( ) = ε E(r,t) + c.c. E 0 (r) 2 exp { iϕ(r) }exp { iωt } after transient damping Complex polarisability α = α + i α (cf. next slide) Force due to the interaction between the field and the induced atomic dipole F = ˆD ε E(r,t) { } ( ) { E + E * } = ε rat 0 αe + α * E * Keeping only the non oscillating terms (time average) F = ε 0 α E E * rat + c.c. 6

12 Radiation pressure and dipole force F = ε 0 α E E * rat + c.c. with E(r,t) = E 0 (r) 2 exp { iϕ(r) }exp { iωt } ( ) + c.c. F = ε α 0 4 E (r) E 0 0 (r) i ϕ(r) E 0 (r) = ε α 0 2 E (r) E 0 0 (r) + ε α 0 E 2 0 (r) ( ) 2 ϕ(r) the expression is evaluated at r = r at F res = ε α 0 2 ( E 0 (r)) 2 ϕ(r) radiation pressure Resonant Dissipative part (imaginary) of polarisability Phase gradient F dip = ε α 0 2 E (r) E 0 0 (r) Dipole force Reactive part (real) of polarisability Amplitude gradient 7

13 Atomic polarisability for a resonance transition E b E a ħω 0 ħω Γ Resonance line: a = ground state Γ a = 0 Line width Γ = Γ b Closed transition Forced dipole oscillation, by E = ε E 0 cos ( ωt ϕ ) = ε ( E + E * ) Density matrix formalism and Optical Bloch equations (dissipation ) complex polarizability α D = ε ˆD ε = ˆD ε 0 d d 0 ˆD ε α = d 2 ε 0 ħ = ε 0 α E + ε 0 α * E * (r,t) 1 (ω 0 ω) i Γ s ħω 1 = de 0 Ω 2 s = 1 /2 (ω 0 ω) 2 +Γ 2 /4 15 linear response saturation 8

14 Radiative forces for a two level atom submitted to a quasi-resonnant laser E(r,t) = E 0 (r) 2 exp { iϕ(r) }exp { iωt } α = d 2 ε 0 ħ 1 (ω 0 ω) i Γ / s = α '+ iα " F = ε α 0 2 E (r) E 0 0 (r) + ε α 0 2 ( E 0 (r)) 2 ϕ(r) r = r at F dip = ε α 0 2 E (r) E 0 0 (r) F res = ε α 0 2 ( E 0 (r)) 2 ϕ(r) Dipole force Resonant radiation pressure Reactive part (real) of polarisability Amplitude gradient Dissipative part (imaginary) of polarisability Phase gradient Substituting α et α by their expressions, we will find properties (and applications) very different for these two forces. 9

15 Resonnant radiation pressure Running wave E(r,t) = E 0 2 exp { ik r}exp { iωt } constant amplitude E 0 F dip = ε α 0 2 E (r) E 0 0 (r) = 0 F res = ε α 0 2 ( E 0 (r)) 2 ϕ(r) = k d 2 2 E 0 2ħ Γ /2 (ω 0 ω) 2 +Γ 2 / s F res =ħk Ω Γ /2 (ω 0 ω) 2 +Γ 2 / s = ħk Γ 2 s 1 + s s = Ω 1 2 /2 (ω 0 ω) 2 +Γ 2 /4 = I I sat (ω 0 ω) 2 / Γ 2 saturation parameter 10

16 Dipole force F dip = ε α 0 2 E (r) E 0 0 (r) with E(r,t) = E (r) 0 exp { iϕ(r) }exp { iωt } 2 F dip = d 2 F dip 0 for an inhomogeneous wave: E 0 0 ħ F dip = ω 0 ω (ω 0 ω) 2 + Γ2 4 2 E 0 (r) 1 + s(r) = ħ(ω 0 ω) 2 U (r) avec U (r) = ħ(ω ω ) 0 2 s(r) 1 + s(r) log 1 + s(r) Derives from a potential (reactive part of polarisability) varying as intensity s(r) = I(r) I sat (ω 0 ω) 2 / Γ 2 Applications atom attracted towards high intensity if ω < ω 0 repelled from high intensity if ω > ω 0 11

17 Dipole trap and optical tweezer U dip (r) = ħ(ω ω 0 ) 2 log 1 + s(r) with s(r) = I(r) 1 I sat 1 + 4(ω 0 ω) 2 / Γ 2 Trapping at the focus of a red detuned laser beam (ω < ω 0 ): optical tweezer laser Shallow trap: demands ultra-cold atoms ( T < 1 mk ) Manipulation of atom clouds, individual atoms, Bose-Einstein atomic condensates, but also of biological objects 12

18 Dipole force: a very useful tool Derives from a potential proportionnal to the light intensity (saturation) U dip (r) = ħ(ω ω 0 ) 2 log 1 + s(r) A remarkable tool to manipulate ultra-cold atoms Optical tweezer Atomic mirror Atomic wave guide Optical lattice ( electrons in a crystal) Disordered medium ( electrons in an amorphous medium) 13

19 Doppler cooling Resonant radiation pressure from two opposite running waves V at ω < ω 0 Non saturating lasers, detuned below resonance Atomic velocity: different Doppler effect for each wave Γ 2 /4 F res =F 0 (ω k V at ω 0 ) 2 +Γ 2 /4 Γ 2 /4 (ω + k V at ω 0 ) 2 +Γ 2 /4 F ω 0 ω k V at Force opposite to velocity: friction cooling ω = ω 0 Γ 2 Generalization to 3-D (6 waves) 14

20 Doppler molasses Near V = 0 : viscous friction F = κ MV or dv dt = κv κ = ħk 2 M s 0 for ω = ω 0 Γ 2 Order of magnitude (rubidium, s 0 = 0.5) κ s 1 Damping time: 40 μs! Exponential cooling! Highly viscous medium: optical molasses atoms stuck d dt V 2 = 2κ V 2 Temperature decreases: where to? Under 1 mk! 15

21 Fluctuations of the resonant radiation pressure Fluctuations of linear momentum P due to spontaneous emission role in resonant radiation pressure. Random jump in P, with magnitude ħk and random direction, at each fluorescence cycle. 2D P = ( ħk ) 2 N fluo coeff. de diffusion (Einstein) On top of the mean force effect, one has a random walk in the P space: d dt P 2 = 2D Heating 17

22 Limits of Doppler cooling Cooling d dt P 2 = 2κ P 2 Heating Steady state P at 2 = D κ Einstein relation Analogous to the evolution of the amplitude of a laser: random walk with elastic force towards equilibrium d dt P = κp at at + F Langevin equation heat Equilibrium temperature Order of magnitude: 100 μk 3 2 k T = P at B 2M = 3 4 ħγ 2 Experimentally observed in 1985 (S. Chu) 18

23 Below the Doppler limit temperature Lowest predicted Doppler temperature: 100 μk range (observed in 1985) Sisyphus cooling : observation (1988) of temperatures in the 10 μk range Interpretation: Sisyphus effect. Takes into account the internal atomic structure (several ground state sublevels) and light polarization. Sub-recoil cooling : 1988: method allowing one to reach residual velocities below the one photon recoil velocity λ de Broglie > λ laser On reaches the nk range ( < 1 μk) : quantum description ψ(r,t) of the atomic motion V R = ħk M Forced evaporative cooling : In a non-dissipative trap (magnetic or dipole), eliminate the atoms with energy > η k B T (typically η 6) then rethermalize by elastic collisions The density in phase space increases => Bose-Einstein condensation! 19

24 Steps of atom cooling 1 c m / s 1 m / s m / s 10 9 K 10 6 K 10 3 K 1 K 10 3 K subrecoil Evaporative cooling and BEC l i q u i d He Doppler molasses Sisyphus room The logarithmic scale emphasizes the magnitude of cooling, characterized by the ratio of the initial to the final temperature, not by the difference. 20

25 Optical tweezer with large numerical aperture lenses Very tightly confined optical dipole trap 810 nm w = 0.8 μm 87 Rb Large Aperture NA = 0.7 Working distance ~ 10 mm 9 spherical lenses Vacuum compatible 780 nm 87 Rb 795 nm P 3/2 P 1/2 e 810 nm 810 nm 30 cm Schlosser et al., Nature 411, 1024 (2001) S 1/2 Trap! g

26 Single atom optical tweezer MOT & Dipole Trap Coils Microscopic dipole trap : Spot size < 1 µm Power < 10 mw Trapping time > 1s Imaging :1µm/pixel CCD Camera or Avalanche Photodiode Filters Vacuum chamber 780 nm fluorescence Impossibl e d'afficher l'image. Votre ordinateu r manque peut-être Dipole trap 810 nm Individual atoms «jumping» in the trap «Collisionnal blockade» : only one atom! N. Schlosser et al, Nature 411, 1024 (2001) PRL 89, (2002)

27 Looking at single atoms «Real-time blinking» of individual atoms going in and out of the microscopic dipole trap Fluorescence rate : 8000 cnts/at/s Trapping time : a few seconds

28 «Collisional blockade» N. Schlosser et al, PRL 89, (2002) FET QIPC Main effect : light - assisted collision loss (due to the MOT light) is huge as soon as there are two atoms in the trap When an atom enters the trap with one atom inside, both atoms escape : either zero or one atom only! Some measured trapped atom parameters : Trap frequencies (P trap = 2 mw) : Trap depth (lightshift) : Trap beam waist (P trap = 2 mw) : Temperature : w radial / 2π = 130 khz w axial / 2π = 30 khz 1 mk / mw w 0 = 0.9 µm < 100 µk

29 Collisional Blockade and Beyond N. Schlosser et al, PRL 89, (2002) Behaviour of the number N of trapped atoms : dn/dt = R 0 - Γ N - β N(N-1) R 0 : loading rate, Γ : one-atom loss (0.2 s -1 ), β : two-atom loss Average number of trapped atoms 10 R 0 β R 0 Γ Loading rate (atoms/sec) Large trap w 0 = 10 µm

30 Collisional Blockade and Beyond N. Schlosser et al, PRL 89, (2002) Behaviour of the number N of trapped atoms : dn/dt = R 0 - Γ N - β N(N-1) R 0 : loading rate, Γ : one-atom loss (0.2 s -1 ), β : two-atom loss Average number of trapped atoms R 0 Γ 0.1 R 0 β 10 4 Loading rate (atoms/sec) Smaller trap w 0 = 4 µm R 0 Γ Average number of trapped atoms Loading rate (atoms/sec) Very small trap w 0 = 0.7 µm R 0 β Collisional blockade : specific behaviour of small (< 4µm) dipole trap!

31 Detecting and "heralding" a single trapped atom Fluorescence induced by the cooling lasers (780 nm) fluorescence (cps/ms atom threshold times (sec) Trapping time in the dark ~ 1-3 s 30 No atom

32 A more compact optical setup : aspherical lens Molded aspherical lenses for laser diode objective Working distance 6 mm, N.A. = nm < λ < 900 nm Typical trap parameters for 850 nm, 5 mw : Depth = 2 mk Transverse oscillation frequency = 135 khz

33 Melles Griot Triplet Optical tweezer setup Laser diode 850 nm Aspherical lens in vacuum CCD, SPCM Filter 780 nm 2.15 μm (Airy function 850 nm, theory = 2.09 μm) Transverse field = ± 30 μm

34 Catching single atoms from an slow atomic beam Sortais et al., PRA 75, (2007) counts/sec/atom on the APD Trapping time in the dark ~ 10 s Fluorescence 780 nm Dipole trap Rb Oven 200 m/s Slowing 200 m / s ~ 1 m / s Dilute atomic cloud T ~ 100 μk - 10 μk

35 Trapping 2 single atoms Dipole trap 1 Asph. Vacuum Cross-section on the CCD camera Dipole trap 2 2μm Waist = 0.9(2) μm 100 ms integration time

36 Single atom temperature measurements C. Tuchendler et al, Phys. Rev. A 78, (2008) * Turn off the trap for a short time Δt, then measure the probability to recapture the atom (average on 100 shots) * Fit with a Monte-Carlo calculation for a given temperature T and trap depth (2.5 mk), assuming a thermal distribution. 31 ± 1 µk 168 ± 6 µk Two examples with and without molasse cooling : excellent fit!

37 Another method : Single Atom Time Of Flight A. Fuhrmanek et al, New Journal of Physics 12, (2010) * Imaging the released atom on an intensified CCD (2 stages MCP + fluo) * Free flight then probe pulse 2 µs, probability to detect the atom : 4.4 % * Averaging! 1 µs TOF : 3400 shots, 150 photons 50 µs TOF : shots, 520 photons

38 Single Atom Time Of Flight : results A. Fuhrmanek et al, New Journal of Physics 12, (2010) 150 ± 16 µk Full line : simulation 20 ± 2 µk including instrumental resolution (1 µm) Dashed line : calculated size of the atom cloud 149 ± 15 µk 19 ± 2 µk Comparison with the release and recapture method : excellent agreement!

39 Dipole trap 1 Moving the atom vacuum lens Dipole trap 2 Tip-tilt platform (x-y) Scale of the motion : a few µm in a few ms OK for a quantum register (coherence time : tens of ms)

40 Motion of a tweezer (background light subtracted) I O Institut d Optique Motion length 40 μm Motion radius 40 μm Movie is 100 stills of 100 ms integration. Atoms are being laser cooled while being trapped & moved

41 Thank you for your attention ( and see you soon! )